Multi-scale prediction of chemo-mechanical properties of concrete materials through asymptotic homogenization

Cement and Concrete Research 128 (2020) 105929

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Multi-scale prediction of chemo-mechanical properties of concrete materials through asymptotic homogenization

T

E. Bosco*, R.J.M.A. Claessens, A.S.J. Suiker Department of the Built Environment, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

ARTICLE INFO

ABSTRACT

Keywords: Multi-scale Multi-physics Chemo-mechanical behaviour Finite element method

In the present contribution, the effective mechanical, diffusive, and chemo-expansive properties of concrete are computed from a multi-scale and multi-physics approach. The distinctive features of the approach are that i) the mechanical and diffusive responses are modelled in a coupled fashion (instead of separately, as is usually done), and that ii) the multi-scale model considers three different scales of observation, which allows for including heterogeneous effects from both the micro- and meso-scales in the effective macro-scale properties of concrete. At the macro-scale, the concrete material is considered as homogeneous, whereas at the meso-scale it consists of particle aggregates embedded in a porous cement paste. At the micro-scale the porous cement paste is described as a two-phase material, composed of a solid cement phase and saturated capillary pores. Adopting a two-level asymptotic homogenization procedure, the effective meso-scale properties of the porous cement paste are computed first, using a unit cell that includes the cement paste and pore characteristics. Subsequently, the obtained meso-scale response of the porous cement paste, together with the aggregate characteristics, defines the material properties of a second unit cell, which is used for calculating the effective macro-scale response of concrete. The distributions of the pores and the aggregates within the unit cells are determined from a uniform, random distribution of points, and their radii are defined from a probability distribution function. The efficacy of the proposed framework is illustrated by studying the effective mesoscopic response of a porous cement paste, which demonstrates the influence of the micro-scale porosity and pore percolation. Next, the effective macroscopic response of concrete is analysed, by considering the influence of the aggregate volume fraction, the mismatches in elastic stiffnesses and diffusivity between the aggregate and the cement paste, and the porosity. The computed effective properties are compared with experimental data from the literature, showing a good agreement.

1. Introduction The process of chemo-mechanical degradation is one of the most relevant concerns for durability of concrete structures. In aggressive chemical environments, chloride ingress may lead to steel reinforcement corrosion [1], sulphates reacting with the hydrated cement paste may form gypsum or ettringite, promoting concrete degradation [2], and the reaction of carbon dioxide with the hydrated cement paste forms calcium carbonates that contribute to both concrete damage and the corrosion of reinforcement [3]. The transport of chemical species responsible for concrete degradation takes place via the solutes filling the pores, in the form of ions (for chloride and sulphate attacks), or a dissolved gas (for carbonation), whereby the coupling with the mechanical response leads to chemically-induced stresses [4]. The mechanical and chemical fields are dependent on the heterogeneous

∗

nature of concrete, as characterized by the morphology of the underlying meso-scale and micro-scale material structures, and the physical features of the capillary pore system [5]. An accurate transfer of this lower-scale information towards the macro-scale level is necessary for adequately simulating the impact of chemo-mechanical degradation of concrete in engineering applications, which can be rigorously accomplished by a combined multi-scale, multi-physics approach. In the literature, several analytical models have been developed that calculate the macroscopic effective elastic moduli of concrete from a meso-structure of a single, circular particle aggregate that is possibly surrounded by an interfacial transition zone (ITZ), and is embedded in a cement matrix [6-8]. These models have also been applied at the lower, micro-scale of concrete, in order to compute the mesoscopic effective elastic moduli of a porous cement paste from the micro-structure of a circular hole embedded in solid, cement paste matrix [9]. In an

Corresponding author. E-mail address: [email protected] (E. Bosco).

https://doi.org/10.1016/j.cemconres.2019.105929 Received 1 March 2019; Received in revised form 12 October 2019; Accepted 24 October 2019 0008-8846/ © 2019 Elsevier Ltd. All rights reserved.

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analogous fashion, the effective diffusivity of concrete [10, 11] and the cement paste [12] have been determined, see the comprehensive review paper [13] for an overview of these models. The analytical expressions derived relate the effective properties directly to micro-scale parameters (i.e., porosity, diffusivity and stiffness of solid cement paste) or meso-scale parameters (i.e., aggregate volume fraction, stiffness and diffusivity of porous cement paste and aggregate). However, these expressions typically do not include information about the spatial distributions of the aggregates and pores, or possible correlations between the characteristics of neighbouring aggregates, and thereby oversimplify the underlying heterogeneous material structures of concrete. In order to improve on this aspect, more recently advanced numerical models have been developed, in which the effective macro-scale properties of concrete are determined from two-dimensional [14, 15] or three-dimensional [16, 17] representations of the concrete mesostructure, using computational homogenization techniques [18]. The size, location and orientation distributions of aggregates can be incorporated from statistical information [19], or from tomography data [20], see also [21] for a thorough review. Computational homogenization has further been applied for determining the effective macroscopic diffusive properties from simulations at the meso-scale [22, 23], or from resolving the mass transport characteristics directlys at the micro-scale [24, 25]. Despite the usefulness of the above studies, it should be mentioned that in most cases the mechanical and the diffusive problems are solved separately, thereby neglecting chemo-mechanical coupling effects, such as the stress induced via restrained chemical expansion or contraction. In addition, the numerical homogenization procedure is often applied at a single level (i.e., either at the meso-scale or at the micro-scale), thereby neglecting that heterogeneities may originate from different scales of observation. In the present contribution, the above-mentioned limitations are overcome by i) computing the effective mechanical, chemo-expansive and diffusive properties of concrete from a combined multi-scale, multiphysics approach and ii) considering three different scales of observation, which allows for including heterogeneous effects from both the micro- and meso-scales in the effective macro-scale properties of concrete. These two distinctive features make the proposed approach novel for cementitious materials. The adopted multi-scale method is based on the principles of asymptotic homogenization [26-28]. blackRecent contributions in the literature have presented asymptotic homogenization frameworks to predict the effective properties of heterogeneous materials in the presence of multiple physical fields, focussing on thermo-mechanical [29, 30], thermo-diffusive [31] thermo-piezoelectric [32, 33] and diffusion-reaction [34, 35] problems. By departing from the approach proposed in [36, 37], in this work, the solution strategy is extended to a multi-physics scheme to account for the mechanical and chemo-diffusive interactions. Asymptotic homogenization allows to represent the heterogeneous concrete as an equivalent homogeneous material, for which the effective material properties are retrieved from the fine-scale response through an averaging procedure based on rigorous mathematical principles. The method starts from formulating the displacement and chemical concentration fields as asymptotic expansions. These expansions are subsequently substituted into the coupled equilibrium and diffusion equations, leading to a set of boundary value problems (the so-called “cell problems”) defined at the fine-scale, whose solution provides the fine-scale fluctuations of the displacement and concentration fields. From these fields, the effective mechanical, diffusive and chemo-expansive properties can be computed. Adopting the finite element method, the cell problems are solved numerically for material structures representative of the fine-scale domain. A two-level asymptotic homogenization framework is used here, consisting of two consecutive steps. In the first step, a unit cell problem with micro-structural information on the solid cement paste and the capillary pores is defined for computing the effective meso-scale properties of the porous cement paste. In the second step, these properties are combined with the meso-scale characteristics of the aggregate

to construct a unit cell problem that provides the effective macro-scale properties of the concrete material. The distributions of the pores and the aggregates within the unit cells are determined from a uniform, random distribution of points, and their radii are defined from a probability distribution function. The size of a unit cell needs to be sufficiently large in order to act as a representative volume element (RVE) that is statistically representative for the effective behaviour of real concrete material [38]. The RVE size is here established from the convergence behaviour of effective properties computed as a statistical average over multiple (i.e., 5 or 10) micro-structural realizations. Note that this typically leads to a smaller RVE size compared to when the effective properties would have been calculated from a single microstructural realization. With this approach, the effective mechanical, diffusive and chemo-expansive properties computed at the macro-scale include the dependency of both micro-structural characteristics (porosity and pore percolation) and meso-structural characteristics (aggregate volume fraction, stiffness and diffusivity contrast between the cement paste and the aggregate). A comparison of the calculated effective properties with experimental data taken from the literature illustrates the predictive capability of the present method. The paper is organized as follows. Section 2 presents a review on asymptotic homogenization and the proposed two-level homogenization procedure. The generation of the micro-structural and mesostructural models used in the numerical simulations is discussed in Section 3. The effective properties of the porous cement paste, computed from asymptotic homogenization at the micro-scale, are discussed in Section 4. The effective properties of concrete, calculated from asymptotic homogenization at the meso-scale, are analysed in Section 5. The final conclusions are given in Section 6. 2. Effective material properties through asymptotic homogenization 2.1. A review on asymptotic homogenization Consider the two-dimensional composite illustrated in Fig. 1. The domain Ω, which refers to the coarse-scale, is associated to the characteristic length ℒ, and is constructed from the periodic repetition of a heterogeneous unit cell Q. The unit cell is defined at the fine-scale and has a square shape with sides of length L = η ℒ. When assuming a strong separation between the coarse-scale and the fine-scale, i.e., η ≪ 1, the field quantities governing the mechanical behaviour (displacement and stress) and diffusive behaviour (concentration and flux) can be considered to be dependent on two variables, namely a slow variable X at the coarse-scale and a fast variable x = X/η at the fine-scale. As shown in Fig. 1, the corresponding reference systems for the coarse- and fine-scales are (X,Y ) and (x,y), respectively, which are characterized by the position vectors X = X eX + Y eY and x = x ex + y ey, respectively, with ei a unit vector along the i-axis of a Cartesian coordinate system. In the absence of body forces, the equilibrium equation is given by

= 0,

(1)

where σ is the Cauchy stress tensor and ∇⋅ indicates the divergence operator. In addition, under steady-state conditions, whereby (possible) chemical reactions are neglected, the diffusion equation can be written as

j = 0,

(2)

with j the mass-flux of the diffusing species. The stress tensor σ(X) and the mass-flux j(X) satisfy the constitutive relations1 1 Note that Eq. (3) is based on the small-strain assumption, whereby the antisymmetric part of the displacement gradient, which relates to a rigid-body rotation, is supposed to be zero under the application of appropriate boundary conditions.

2

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Coarse-scale Fine-scale (b)

(b)

(b)

(b)

(b)

(b)

(b)

(b)

(b)

=

ℒ

Ω (b)

(b)

(b)

(b)

= (b)

(b)

(b)

(b)

ℒ

Fig. 1. Coarse-scale domain Ω and underlying periodic fine-scale unit cell Q.

(X) = 4C (x): ( u (X) j (X ) =

D (x):

• The first mathematical problem allows to calculate the first-order

(3)

(x) c (X))

influence functions 3N1(x), b1(x) and m1(x) related to the first-order terms u1 and c1 in the asymptotic expansions (7) and (8), respectively. Here, the functions 3N1(x) and b1(x) represent the periodic fluctuations of the displacement field within the fine-scale material structure, due to a variation of the average coarse-scale strain and concentration fields, respectively. Additionally, m1(x) represents the periodic fluctuation of the concentration field at the fine-scale, associated to the variation of the coarse-scale concentration gradient. The functions 3N1(x), b1(x) and m1(x) are obtained as the periodic solutions of the following unit cell problems, see [36] for more details:

(4)

c (X),

with ∇ the gradient operator, u(X) the displacement field and c(X) the concentration field. The response of the coarse-scale domain depends on the material properties of the underlying fine-scale structure, and thus is described by local constitutive tensors that are periodic functions of the fast variable x. As illustrated in Eq. (3), in the present study the relevant constitutive tensors are the fourth-order elasticity tensor 4 C(x), the second-order expansion tensor β(x), and the second-order diffusivity tensor D(x). Furthermore, β(x)Δc(X) represents the expansive strain due to a concentration change Δc = c − c0 of the diffusing species, with c0 being the reference value. This term accounts for the coupling between the mechanical and diffusional fields. Substituting the constitutive relations (3) and (4) into (1) and (2), respectively, results in a set of differential equations as a function of the displacement and concentration fields at the fine-scale level:

( 4C (x):

(D (x):

u (X))

( 4C (x):

(x) c (X)) = 0

(5)

c (X) = c0 (X , X/ ) + c1 (X , X / ) +

2 u (X , 2

2 c (X , 2

X / ) + ...

X / ) + ...

( 4C (x): (

b1 (x)

+ 4IS )) =3 0

(9)

(x))) = 0

(10) (11)

4 S

where I is the fourth-order symmetric identity tensor, defined as S Iijkl = ( il jk + ik jl )/2 , and I is the second-order identity tensor, Iij = δij, with δij the Kronecker delta symbol. Eqs. (9)– (11) need to be satisfied in each material point of the fine-scale domain, whereby the solution is dependent on the fine-scale constitutive properties 4 C(x), β(x) and D(x). The problem formulation is completed with the definition of periodic boundary conditions expressed in terms of the (unknown) fields 3N1(x), b1(x) and m1(x), requiring that the values assumed by each influence function at corresponding points on opposite boundaries (left and right/top and bottom) remain constant during the simulation. Moreover, for warranting the uniqueness of the solution, the influence functions 3N1(x), b1(x) and m1(x) are enforced to vanish on average on the fine-scale domain:

Two important aspects that need to be mentioned here are that the computational solution of Eqs. (5)– (6) over the full coarse-scale domain Ω is relatively expensive, and that the obtained values of the local fine-scale displacement and concentration fields are generally too detailed to be used in practice. These aspects can be improved by computing a solution based on asymptotic homogenization. Asymptotic homogenization allows to replace the heterogeneous medium with rapidly oscillating material properties 4C(x), β(x) and D(x) by an equivalent homogeneous domain, which is characterized by effective material properties calculated from the fine-scale properties through an averaging procedure [26-28]. Asymptotic homogenization assumes that the displacement field u(X) and concentration field c(X) can be developed via an asymptotic expansion in terms of η, i.e.,

u (X) = u 0 (X , X / ) + u1 (X , X / ) +

3 N (x) 1

(D (x) ( m1 (x) + I)) = 0,

(6)

c (X)) = 0.

( 4C (x): (

1 |Q|

(7) (8)

1 |Q|

The averaging procedure departs from substituting expressions (7) and (8) into Eqs. (5) and (6), respectively, leading to the definition of two sets of mathematical problems described below. 3

3

N1 (x)dQ =3 0 Q

Q

b1 (x)dQ = 0

(12)

(13)

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1 |Q|

•

Q

m1(x)dQ = 0

.

shown in Fig. 2 (b). In the literature, the presence of an interfacial transition zone (ITZ) at the interface between the aggregates and the cement paste is reported [17, 40], which is characterized by a higher porosity than that of the adjacent cement paste. The ITZ thickness is typically small (10–50 μm) compared to the characteristic size of an aggregate (10–40 mm). Moreover, analyses of experimental data [13] and numerical simulations [5] on the diffusivity of cement mortar and concrete only show a minimal influence of the ITZ on the effective diffusive properties. For these reasons, in this work the contribution of the ITZ has been neglected. The micro-scale of concrete, which is associated to the characteristic length ℓ, is represented by the porous cement paste illustrated in Fig. 2 (c). The porous cement paste is composed of a solid phase and capillary pores that may be filled with water. The solid phase typically consists of amorphous calcium silicate hydrate, crystalline calcium hydroxide, and unhydrated cement products. Due to the difficulty of identifying material properties for each of these individual components, the solid phase here is treated as an isotropic, homogeneous medium, as is often done in the literature [10, 12, 41]. However, the homogenization approach can be straightforwardly extended to account for the different solid phases at the micro-level, by modelling them as separate components, e.g., along the lines of [25]. Note also that the model allows for the incorporation of even smaller length scales, for instance, to explore the influence on the effective material properties by heterogeneities in the calcium silicate hydrate phase [42].

(14)

Note that similar unit cell problems can be defined to calculate the second-order contributions related to terms u2 and c2 in the asymptotic expansions (7) and (8). Higher-order contributions become relevant if the wavelengths of the mechanical and diffusional fields induced reach a similar magnitude as the size of the heterogeneities within the unit cell. This, however, is not the case for the problems considered in this work; the mechanical and diffusional fields generated slowly vary across the size L = η ℒ of the unit cell. In the second mathematical problem, the effective properties at the coarse-scale, which are the effective elastic stiffness 4C , the effective expansion tensor and the effective diffusion tensor D , are computed from corresponding, average responses in the fine-scale domain as [36] 4C

=

=

D =

1 |Q|

Q

4C (x):

1 4¯ 1 C : |Q|

1 |Q|

Q

Q

(

3 N (x) 1

4C (x):

( (x)

+ 4IS )dQ

(15)

b1 (x))dQ

D (x): ( m1 (x) + I)dQ

(16)

.

(17)

2.2.1. Homogenization between the micro-scale and meso-scale Since the asymptotic homogenization procedure includes three different scales of observation, see Fig. 2, it needs to be carried out in two consecutive steps. In the first homogenization step the effective properties of the cement paste are computed at the meso-scale from asymptotic homogenization of the micro-scale behaviour. Applying the terminology introduced in Section 2.1, the meso-scale should thus be interpreted as the coarse-scale, and the micro-scale as the fine-scale. At the meso-scale the cement paste is considered as homogeneous, while at the micro-scale the presence of saturated pores is modelled explicitly via the micro-structure depicted in Fig. 2 (c). The material properties of the solid phase and the pore phase of the cement paste are indicated by the subscripts “c” and “ϕ”, respectively. The asymptotic homogenization procedure departs from generating micro-structural geometries characterized by different values of the porosity, as described in detail in Section 3. Next, the unit cell problems, Eqs. (9) to (11), are solved for these micro-structures, which provides the values of the influence functions. These values are subsequently substituted into relations (15)

In specific, the effective material properties , and D are obtained by first solving the unit cell problems (9)– (11) for a given fine-scale geometry, followed by substituting the result into Eqs. (15)– (17). For more details on the derivation of the effective properties, the solution of the cell problems, and the numerical implementation, the reader is referred to [26, 31, 36, 37] . 4C

2.2. Two-level homogenization procedure Concrete is a multi-phase material characterized by a complex heterogeneous structure that spans across different length scales, ranging from the nanometre to the decimetre [39]. For the analysis of its mechanical and diffusive properties, three main length scales can be distinguished, as depicted in Fig. 2. At the macro-scale, which is associated to the characteristic length ℒ, concrete is assumed to be a homogeneous medium characterized by effective material properties, see Fig. 2 (a). At the meso-scale, which is related to the length scale L, concrete is idealized as a two-phase material composed of particle aggregates embedded in the (porous) cement paste, as schematically

Macro-scale

Meso-scale

¯

to (17) to calculate the effective properties of the cement paste, 4Cc , ¯

¯

c

and Dc .

Micro-scale

ℒ

ℓ

(b)

• Homogeneous material

• Porous cement paste • Aggregates

• Cement paste • Saturated pores

(a)

(b)

(c) 4

Fig. 2. Idealized representation of concrete involving three different length scales: (a) the macro-scale with characteristic size ℒ, (b) the meso-scale with characteristic size L and (c) the micro-scale with characteristic size ℓ. As illustrated in the figure, for computing the effective material properties at the macro-scale a twolevel homogenization procedure is applied.

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2.2.2. Homogenization between the meso-scale and macro-scale In the second homogenization step the effective properties of the concrete are computed at the macro-scale from asymptotic homogenization of the meso-scale behaviour. The macro-scale is thus interpreted as the coarse-scale, and the meso-scale as the fine-scale. In accordance with Fig. 2 (b), the mesoscopic structure is assumed to be a heterogeneous medium composed of cement paste and particle aggregates. The cement paste is characterized by the porosity-dependent ¯

¯

placed first in the meso-structural domain, followed by the smaller particles, whereby particle overlap is prevented by means of a dedicated numerical algorithm. Preliminary analyses have shown that the specific particle size distribution chosen does not significantly influence the effective properties computed, in a sense that the results found for mono-disperse and poly-disperse particle assemblies are similar. In the present work the particle assembly is chosen as poly-disperse, which allows for investigating unit cells with relatively high values of the aggregate volume fraction, up to v ≈ 0.75. For the application of the asymptotic homogenization framework, (geometrical) periodicity of the small scale unit cell must be satisfied, see also Fig. 1. To this aim, particles partially falling outside the unit cell (0,L) × (0,L) are cut along the specific cell boundary, and copied at the opposite cell boundary. In addition, the periodic unit cell needs to be taken sufficiently large in order to be representative of a real stochastic, heterogeneous concrete micro-structure [38]. The determination of the actual size of the meso-scale representative volume element (RVE) is discussed in detail in Section 5.1. An automatic and efficient meshing procedure was developed, whereby the unit cell was first discretized by a dense regular grid of square finite elements of edge length le = L/ξ, with the integer ξ ≥ 1. For each element e it was checked whether it is part of a particle i, which is the case if its geometrical centre is located inside the particle domain, as defined by the circular area ri2 centred at X ic . Note that this procedure results in a zigzag shape of the particle boundaries, which has an influence on the values computed for the local stresses and strains. However, it has been confirmed by a mesh convergence study that the influence on the effective properties is negligible if the parameter ξ is chosen sufficiently large. As an illustrative example, Fig. 3 (a) depicts a meso-scale geometry generated for a volume fraction v = 0.5, whereby the mean value and standard deviation of the particle radius are r¯ = 0.02L and rσ = 0.01L, respectively, with L the side length of the square unit cell. A detail of the domain and the corresponding finite element discretization, in which ξ = 400, are shown in Fig. 3 (b) and (c), respectively. The generation of the micro-scale unit cell is performed along similar lines as described above for the meso-scale unit cell. The square micro-scale domain with sides of length ℓ = ηL consists of a solid cement paste with embedded capillary pores. The porosity ϕ is represented by the ratio between the total pore volume (per unit depth) and the volume of the unit cell. All pores are assumed to have the same radius. The centre locations of the pores are obtained from a uniform random distribution, and pore overlapping is allowed to occur in the pore deposition procedure. The generated micro-structure is extended periodically, in the same fashion as described above for the mesostructural domain. The micro-scale unit cell is required to be an RVE, as elaborated in Section 4.2. Finally, the meshing procedure is similar as for the meso-scale domain.

¯

effective properties 4Cc , c and Dc computed in the micro-meso homogenization step outlined above, while the material properties of the aggregates are described by the constitutive tensors 4Ca, βa and Da. Meso-structural geometries are generated for a range of aggregate volume fractions, for which the unit cell problems, Eqs. (9)– (11), are solved. The obtained influence functions are used to compute the effective material properties 4C , and D at the macro-scale in accordance with Eqs. (15) to (17). Accordingly, with the current two-level homogenization procedure the effective constitutive properties calculated at the macro-scale include both meso-structural and micro-structural information. 3. Meso-scale and micro-scale models The micro-scale and the meso-scale unit cells used for computing the corresponding effective properties show strong similarities in the way they are constructed. As illustrated in Fig. 2 (b), the meso-scale unit cell is modelled as a square heterogeneous domain, which, in accordance with Fig. 1, has dimensions L × L, and consists of particle aggregates that are embedded within a cement paste. The aggregate volume fraction v of the meso-scale configuration is initially prescribed, and is specified as the ratio between the total volume (per unit depth) of the particle aggregates, Va, and the volume of the unit cell |Q|:

v=

Va . |Q|

(18)

The aggregates are generated by assuming a circular shape and defining a set of particle radii in accordance with the standard normal probability density function:

f (r|¯, r r)=

1 e r 2

(r r¯)2 2r 2

.

(19)

In this equation, the parameter r¯ is the mean particle radius, which may be interpreted as a characteristic length of the particle aggregate, while rσ is the standard deviation of the distribution of particle radii. The location of a particle in the domain Q of the unit cell is determined by its geometrical centre Xc, which is taken from a uniform, random distribution of points. The generated particle radii are sorted from the largest to the smallest value; the particles with the largest radii are

(a)

(b)

(c)

Fig. 3. (a) Example of a meso-scale configuration with an aggregate volume fraction v = 0.5; (b) detail of the region indicated in (a) by the square and (c) corresponding finite element discretization.

5

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4. Effective properties of porous cement paste

as a/b = [1,5,10,15,20]. Note that for an increasing aspect ratio the pore percolation increases through the formation of channels that promote diffusion within the cement micro-structure. The unit cells are discretized using bilinear quadrilateral finite elements equipped with a four-point Gauss quadrature. As mentioned, the finite elements are square-shaped and have a size le = ℓ/ξ, with ξ = 400. This results in a regular grid of 400×400 elements, which has been confirmed to be sufficiently fine for providing mesh-independent results. Additional numerical simulations, not presented here, were performed to validate the proposed asymptotic homogenization scheme. A micro-scale reference model, characterized by a porosity ϕ = 0.3, was generated and its effective properties were computed by means of asymptotic homogenization. The same micro-scale geometry was subsequently used to carry out a hygro-elastic and a steady-state diffusive finite element analysis, where periodic boundary conditions for the displacement and concentration fields were applied. From these simulations, the effective properties were computed by means of computational homogenization. The results of the two methods turn out to be in excellent agreement, with a relative difference in the calculated effective properties lower than 0.1%. In the presentation of the numerical results, for generality the effective elastic modulus c and the effective expansion coefficient ¯c of the meso-scale cement paste are considered relative to the corresponding properties of the solid cement phase at the micro-scale, Ec and βc. Further, the effective diffusion coefficient D¯c is normalized with respect to the diffusion coefficient of the pore phase Dϕ.

4.1. Input parameters for the simulations In the micro-structural simulations chloride ions are assumed to be the reference diffusing species. Note, however, that without loss of generality the approach is also directly applicable to other chemical species. The solid cement paste is modelled as a homogeneous, isotropic material, characterized by the Young's modulus Ec, Poisson's ratio νc, expansion coefficient βc, and diffusion coefficient Dc. In all simulations, the elastic modulus and the Poisson's ratio of the solid cement paste are taken as Ec = 45.35 MPa [9] and νc = 0.2 [43], respectively. It has been experimentally observed that the expansion of hardened cement in chloride solutions is larger than that in distilled water [4, 44], although specific values of the expansion coefficient of the cement paste are not available in the literature. Hence, for simplicity, the expansion coefficient of the cement paste containing a chloride solution in the pores is taken equal to that of a cement paste containing a water solution, i.e., βc = 2.46 ⋅ 10−6 [45]. The porous phase is considered to be fully saturated with water, whereby the influence of water on the effective mechanical and expansive responses of the cement paste is neglected, i.e., the elastic and expansive properties of the pores are assumed to be equal to zero, Eϕ = 0, νϕ = 0 and βϕ = 0. The diffusive behaviour of the chloride ions in the pores is supposed to be isotropic, whereby the diffusion coefficient equals Dϕ = 2.032 ⋅ 10−9 m2/s [11]. The diffusion coefficient of the solid paste is varied with respect to the diffusion coefficient of the porous phase by selecting three different values, Dc/ Dϕ = [10−3,2.5 ⋅ 10−3,10−2]. Here, the intermediate value Dc/ Dϕ = 2.5 ⋅ 10−3 is based on the empirical formula proposed in [24], which has also been applied in other numerical studies [25]. The minimum size of the unit cell for representing an RVE is determined by varying the cell edge ℓ with respect to the pore radius r, i.e., ℓ/r = [5,25,50,75,100]. The influence of the porosity on the effective response of the micro-structure is investigated by varying the porosity in the range ϕ = [0 − 0.3], which is a realistic range for common cement pastes [46]. The effect of pore percolation on the calculated effective properties is analysed by considering micro-structures of the same porosity (ϕ = 0.3), but with different pore connectivity. Here, the pores are assumed to have an elliptical shape, whereby the aspect ratio between their long a and short b axes is varied

4.2. Definition of an RVE for the porous cement paste In order to obtain statistical representative information from the micro-structural analyses, the minimum size of the RVE should be defined first. The RVE size is established from the convergence behaviour of effective properties computed as a statistical average over multiple (5 or 10) micro-structural realizations. Essentially, the minimum RVE size corresponds to the specific ratio between the cell size ℓ and the pore radius r at which the effective diffusive and mechanical properties have reached constant values within a certain inaccuracy. The dimensionless effective elastic modulus E¯c / Ec and the dimensionless diffusion coefficient D¯c / D are, respectively, shown in Fig. 4 (a) and (b) as a function of the ratio ℓ/r. The numerical results are computed for the lowest of the

Fig. 4. Effective properties as a function of the relative cell size ℓ/r, for three different porosities, ϕ = 0.1, 0.2 and 0.3. (a) The dimensionless elastic modulus E¯c / Ec and (b) the dimensionless diffusion coefficient D¯c / Dc of the porous cement paste. The diffusivities of the solid cement paste and the pores have been selected in accordance with Dc/Dϕ = 10−3. The symbols indicate the properties computed from 5 individual micro-structural realizations, and the lines connect the corresponding average values. 6

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three diffusivity ratios selected, Dc/Dϕ = 10−3, which corresponds to the largest contrast in the diffusion coefficients of the constituents, and therefore is expected to lead to the largest (most critical) minimum size of the RVE. The symbols in Fig. 4 indicate the properties calculated from 5 individual micro-structural realizations, whereas the lines represent the average values. The porosities analysed are ϕ = 0.1 (triangles, dashed line), ϕ = 0.2 (squares, dotted-dashed line) and ϕ = 0.3 (circles, dotted line). The values of the effective properties are calculated as an average of the values measured in x − and y − directions, which reduces the influence of possible anisotropy due to the randomness of the pore distribution in the micro-structure. It can be observed that all curves asymptotically converge to an approximately constant value for an increasing relative cell size ℓ/r. The representative value found for the effective elastic modulus decreases under increasing porosity, while the value calculated for the effective diffusivity increases with porosity. It is further illustrated that the scatter in values under different micro-structural realizations tends to decrease for increasing relative cell size ℓ/r, which is due to the fact that a larger unit cell better accounts for statistical micro-structural variations. In order to define the minimal RVE size, the effective properties are considered to be converged if the average value computed has a relative difference smaller than 5% of the value calculated for the largest unit cell, ℓ/ r = 100. Note that the convergence towards an RVE typically goes faster if the porosity is smaller, as a result of a lower scatter in the computed effective properties. Nonetheless, for all porosities considered an RVE is obtained when ℓ = 50r, which thus is the unit cell size selected for the simulations presented further in this communication. Note that in Fig. 4 the effective expansion has not been considered. This is, since the pores allow for a free expansion of the adjacent cement paste, by which the effective expansivity of the micro-structure becomes equal to the expansivity of the solid cement paste for an arbitrary size of the unit cell.

with asymptotic homogenization, as obtained from averaging over 5 micro-structural realizations, are depicted together with results obtained from other homogenization methods and experimental data reported in the literature. The experimental data for the effective elastic modulus of the cement paste were taken from [47], and were normalized by using the elastic modulus of the solid cement paste of Ec = 45.35 MPa reported in [9]. Fig. 5 (a) illustrates that the effective elastic modulus generally decreases with increasing micro-structural porosity, whereby the Voigt upper bound (dotted line) and the HashinShtrikman upper estimate (dashed line) clearly capture the experimental data less accurately than the asymptotic homogenization method (solid line). Note that lower bounds and estimates of the effective elastic modulus are not shown here, because these are governed by the properties of the compliant pore phase, and for ϕ > 0 thus are equal to zero. The effective diffusion coefficient of the porous cement paste is depicted in Fig. 5 (b) for three selected diffusivity ratios, Dc/ Dϕ = [10−3,2.5 ⋅ 10−3,10−2]. It can be observed that the effective diffusion coefficient increases with porosity, which, in general, is also what the experimental data shows. The experimental data of the porous cement paste have been reported in [48-51], and have been subsequently summarized in [11]. In Fig. 5 (b), these data are normalized with respect to the value of the pore diffusivity, Dϕ = 2.032 ⋅ 10−9 m2/ s [11]. For a relatively low porosity 0 ≤ ϕ < 0.2, the experimental data are comparable to the result of asymptotic homogenization obtained for Dc/Dϕ = 10−3. For a larger porosity ϕ = 0.2, the experimental data more closely match the asymptotic homogenization result for Dc/ Dϕ = 2.5 ⋅ 10−3. The selected ratio of Dc/Dϕ = 10−2, however, seems to overestimate the experimental data, and therefore will not be considered in the forthcoming analyses. It can be also observed that the experimental data on average show a relatively strong increase of the diffusion coefficient at larger porosity, ϕ = 0.2. This effect may be due to pore percolation, which creates diffusion channels in which the transport of ions is enhanced, resulting in a higher effective diffusivity. This aspect will be analysed in more detail in the next section. For comparison, the Reuss lower bound (dotted line) is also shown in Fig. 5 (b), which, under growing porosity, becomes increasingly different from the result computed by asymptotic homogenization. The

4.3. Influence of porosity on the effective properties Fig. 5 (a) and (b) respectively show the effective dimensionless elastic stiffness and diffusion coefficient of the cement paste as a function of the porosity. The effective material properties calculated

Fig. 5. Effective properties as a function of the porosity ϕ. (a) The dimensionless elastic modulus E¯c / Ec and (b) the dimensionless diffusion coefficient D¯c / Dc of the porous cement paste. The diffusivities of the solid cement paste and the pores have been selected in accordance with Dc/Dϕ = [10−3,2.5 ⋅ 10−3,10−2]. The results obtained by asymptotic homogenization are compared to classical bounds and estimates and experimental data reported in the literature [47-51]. 7

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Fig. 6. SEM micrographs of (copper slag reinforced) concrete, showing: (a)–(b) capillary pores of circular shape and (c)–(d) a capillary pore channel. Image elaborated from [52].

Hashin-Shtrikman lower estimate virtually coincides with the Reuss lower bound, and therefore is not presented here. Also, the Voigt upper bound and Hashin-Shtrikman upper estimate are not shown, since these are strongly governed by the diffusive properties of the pore phase, which are much larger than those of the cement paste and therefore provide unrealistic results.

characterized by pores of both circular and elongated shape. The exact distribution of the different pore geometries and their relative dimensions are typically uncertain, which makes it difficult to define and generate these micro-structures. The adopted idealized models with pores of constant shape therefore are an appealing alternative to investigate the influence of percolation on the effective material properties. Here, it should be mentioned that the two-dimensional nature of the studied micro-structures is expected to enhance the effect of pore percolation. Hence, the present analysis results on the influence of pore percolation on the effective properties should be interpreted as purely qualitative. The normalized effective elastic modulus E¯c / Ec and diffusion coefficient D¯c / D of the porous cement paste are respectively shown in Fig. 8 (a) and (b) as a function of the pore aspect ratio a/b. The effective properties are computed as the average of 10 micro-structural realizations; this number has been determined from a preliminary study, which indicated that the use of 5 micro-structural realizations occasionally may be insufficient for obtaining a statistically isotropic result if the pores are strongly elliptic. From Fig. 8 (a) it can be observed that the effective elastic modulus of the porous cement paste drastically decreases under increasing pore aspect ratio, whereby it virtually becomes zero at larger pore aspect ratios a/b ≥ 10, due to percolation channels crossing the unit cell, see Fig. 7 (b) –(c). The effective diffusion coefficient shown in Fig. 8 (b) is computed for two relevant diffusivity ratios, Dc/Dϕ = [10−3,2.5 ⋅ 10−3]. Apparently, for both diffusivity ratios the increase in the effective diffusion coefficient under increasing pore aspect ratio is relatively mild for aspect ratios 0 ≤ a/b < 10, but increases substantially for higher aspect ratios a/b ≥ 10. This is due to the promotion of fast ion diffusion by a fully percolated micro-structure. In Section 5 the effective properties will be computed at the macroscale, through applying the asymptotic homogenization procedure to the meso-scale structure composed of porous cement paste and

4.4. Influence of pore percolation on the effective properties In the micro-scale model, the capillary pores are assumed to have a circular shape. This assumption was also made in other modelling studies, e.g., [9], and is motivated from experimental observations of concrete micro-structures. Fig. 6 (a) shows a scanning electron microscope (SEM) image of a section of (copper slag reinforced) concrete, whereby the dark circles visible in the detailed magnification in Fig. 6 (b) are the capillary pores. Additional SEM images of concrete microstructures, however, reveal that the capillary pores may also appear via more elongated shapes that can create percolation channels, see Fig. 6 (c) and the detailed magnification in Fig. 6 (d). This is particularly relevant for porosity values ϕ > 0.18, at which a percolation threshold can be identified [5, 24]. In the present study, the effect of pore percolation on the mechanical and diffusive properties of the cement paste is examined as follows. At a fixed level of porosity, ϕ = 0.3, various micro-structures are generated in which the pores are modelled as ellipses, thereby considering a range of aspect ratios, a/b = [1,5,10,15,20]. The elliptical pores are oriented along angles following from a random distribution. For the micro-structure with circular pores (a/b = 1) shown in Fig. 7 (a), percolation channels clearly remain absent. Conversely, for the microstructure with elliptical pores with aspect ratios of a/b = 10 and a/ b = 20, the pore geometry allows for establishing percolated connectivities across the size of the unit cell, see Fig. 7 (b) and (c), respectively. Note, however, that a realistic micro-scale domain is

Fig. 7. Various cement paste micro-structures with a porosity of ϕ = 0.3, used to investigate the effect of pore percolation on the effective material properties. The pore connectivity depends on the specific pore aspect ratio a/b, as illustrated for: (a) circular pores with a/b = 1; (b) elliptical pores with a/b = 10; (c) elliptical pores with a/b = 20.

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Fig. 8. Effective properties as a function of the pore aspect ratioa/b for a porosity ϕ = 0.3. (a) The dimensionless elastic modulus E¯c / Ec and (b) the dimensionless diffusion coefficient D¯c / Dc of the porous cement paste.

aggregates, see Fig. 2. For simplicity, the effect of pore percolation is omitted in this analysis, by assuming the micro-structural pores to be circular.

may be observed, depending on the aggregate type [54]. Hence, for a specific value of the aggregate volume fraction v = 0.5, the effective properties are calculated for a broad range of stiffness and diffusivity ratios, Ea/Ec and Da/Dc =[10−4,10−2,10−1,0.5,1,5,101,102,104]. The finite element discretization used for the meso-structure is similar as for the micro-structure, and is constructed with bilinear quadrilateral finite elements equipped with a four-point Gauss quadrature. The elements are square-shaped and have a size le = L/ξ with ξ = 400, whereby it was confirmed that this discretization is sufficiently fine to provide mesh-independent results.

5. Effective properties of concrete 5.1. Input parameters for the simulations At the meso-structural level, the porous, isotropic cement paste is characterized by the elastic modulus E¯c , Poisson's ratio ¯c , expansion ¯

coefficient c and diffusion coefficient D¯c . The dependency of these properties on the micro-scale porosity has been computed in Section 4 in the homogenization step performed between the microscopic and mesoscopic scales. The material behaviour of the particle aggregates is also considered as isotropic, with the elastic modulus, Poisson's ratio, expansion coefficient and diffusion coefficient given by Ea, νa, βa, Da, respectively. The influence of the aggregate volume fraction on the effective macroscopic properties of concrete is investigated in Section 5.2. To this aim, a series of meso-structural geometries is considered, whereby the aggregate volume fraction v is varied in the range [0 − 0.75]. The elastic modulus and Poisson's ratio of the aggregate are assumed to be representative of sand, in accordance with Ea = 4Ec and νa = 0.2, respectively [53]. Such a composition may be interpreted as a “fine-grained concrete”; further in the analysis aggregates with higher relative stiffness will be considered as well, as representative of coarse grains (i.e., gravel) present in concrete materials. The diffusivity of the aggregate typically is much lower than the diffusivity of the cement paste [10], such that it can be taken as zero, Da = 0. In addition, the expansion coefficient of the particle aggregates can also be considered to be ignorable with respect to that of the cement paste, yielding βa = 0. The minimum size of the RVE at the meso-scale level was established in the same way as at the micro-scale level, where the effective macroscopic properties were determined by homogenizing the responses obtained from 5 meso-structural realizations. The convergence study was performed using the stiffness mismatch of Ea = 4Ec mentioned above, which resulted in a mesoscopic RVE with side length L = 50¯r , where r¯ is the average radius of the aggregates. The influence of the mismatch between the material properties of the aggregate and the porous cement paste is studied in Section 5.3. Although the values of the elastic and diffusive properties mentioned above are realistic for certain types of aggregates i.e., sand, the material parameters of the aggregate may in general vary over a substantial range. For instance, a significant variation in the diffusive properties

5.2. Effect of aggregate volume fraction The effect of the aggregate volume fraction on the effective mechanical, chemo-expansive and diffusive properties of a fine-grained concrete is presented in Fig. 9. The results of the numerical simulations are depicted using the dimensionless effective elastic modulus / Ec , expansion coefficient ¯/ c and diffusion coefficient D¯c / D . The effective properties of the porous cement paste, calculated in Section 4 as a function of porosity with ϕ = [0 − 0.3], have been used as input for the simulations, adopting the ratio Dc/Dϕ = 10−3 between the diffusivities of the solid cement paste and the capillary pores. Fig. 9 (a) shows that the effective elastic modulus E increasingly grows when the volume fraction v of the aggregates becomes larger, whereby the amount of growth is approximately independent of the porosity of the cement paste. Note that at increasing porosity ϕ from 0 to 0.3 the effective elastic modulus typically decreases by a factor between 1.5 and 2.5. The obtained numerical results are compared with experimental data for a cement mortar containing a sand aggregate, taken from [53]. Comparable to the modelling results, the measured elastic moduli are larger for a higher aggregate volume fraction, although the magnitude of this increase appears to be somewhat less. This difference may be ascribed to uncertainties in the definition of the material properties for the individual phases, to the lack of information on the exact value of the porosity in the cement mortar tested, and to the fact that the numerical simulations are a two-dimensional representation of the three-dimensional experimental samples. Fig. 9 (b) illustrates that the effective expansion coefficient decreases with increasing aggregate volume fraction. This decrease can be explained from the fact that the amount of swelling and shrinkage of the cement paste typically is constrained by the presence of relatively stiff aggregates. Conversely, the pores do not offer mechanical resistance to volumetric changes of the cement paste, as a result of which 9

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Fig. 9. Effective properties of concrete as a function of the aggregate volume fraction v, for various values of the cement paste porosity, ϕ = [0,0.1,0.2,0.3]. (a) Dimensionless elastic modulus / Ec , with experimental data taken from [53], (b) dimensionless expansion coefficient ¯/ c , and (c) dimensionless diffusion coefficient D¯ / D , with experimental data taken from [55]. The diffusivity of the porous cement paste is taken in accordance with Dc/Dϕ = 10−3 and the diffusivity of the aggregates is assumed to be negligible, Da = 0 [10].

the effective expansion is virtually independent of the porosity ϕ. Fig. 9 (c) finally shows the effective diffusion coefficients of the finegrained concrete as a function of the aggregate volume fraction, plotted on a semi-logarithmic scale. It can be observed that the effective diffusivity decreases rapidly with increasing aggregate volume fraction, since the diffusivity of the aggregates is much smaller than that of the cement paste. The computational results are compared to experimental data taken from [55], referred to as mortars with sand aggregates, which apparently follow a more or less similar trend. Furthermore, under increasing porosity the effective diffusivity becomes larger, as the diffusion coefficient of the pores is substantially larger than that of the cement paste, in accordance with Dc/Dϕ = 10−3.

The effective expansion coefficient depicted in Fig. 10 (b) decreases under an increasing stiffness mismatch Ea/Ec, whereby the largest variation again occurs in the range 10−2 < Ea/Ec < 102. Essentially, a stiffer aggregate more strongly constrains the cement paste, thereby reducing the overall expansion of the meso-structure. As in Fig. 9 (b), the effect of the porosity on the effective chemical expansion is small. In the logarithmic plot of Fig. 10 (c) the dimensionless effective diffusion coefficient D / D is depicted as a function of the ratio between the diffusion coefficients Da/Dc for selected values of the porosity ϕ = [0,0.1,0.2,0.3] and a reference aggregate volume fraction v = 0.5. For all porosity values the effective diffusivity grows under an increasing diffusivity mismatch Da/Dc, which is most significant in the range 10−2 < Da/Dc < 102. Outside this interval, the effective diffusivity changes only mildly.

5.3. Influence of stiffness and diffusivity mismatches Fig. 10 (a) and (b) present the dimensionless effective elastic

6. Conclusions

modulus E / Ec and expansive coefficient / c as a function of the stiffness mismatch Ea/Ec between the aggregate phase and the solid cement phase, plotted on a semi-logarithmic scale for selected values of the porosity ϕ = [0,0.1,0.2,0.3] and a reference aggregate volume fraction v = 0.5. The effective elastic modulus plotted in Fig. 10 (a) increases with increasing stiffness mismatch Ea/Ec, independent of the value of the porosity. This increase is the strongest in the range 10−2 < Ea/Ec < 102; outside this range the effective elastic modulus changes only mildly, and approaches an asymptotic value.

The focus of this work is on the prediction of the effective mechanical, chemo-expansive and diffusive properties of concrete by means of a combined multi-scale and multi-physics approach, which is highly relevant for assessing its material behaviour under coupled chemo-mechanical loading conditions. The complex heterogeneous structure of concrete is studied by distinguishing three different scales of observation: a homogeneous macro-scale, a heterogeneous mesoscale consisting of particle aggregates embedded in the porous cement 10

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Fig. 10. Effective properties of cement mortar as a function of the mismatch in elastic stiffness Ea/Ec (a) and (b) and diffusivity Da/Dc (c), for selected values of the cement paste porosity, ϕ = [0,0.1,0.2,0.3] and an aggregate volume fraction v = 0.5. (a) Dimensionless elastic modulus / Ec , (b) dimensionless expansion coefficient ¯/ , and (c) dimensionless diffusion coefficient D¯ / D . The diffusivity of the porous cement paste is taken in accordance with Dc/Dϕ = 10−3. c

paste, and a heterogeneous micro-scale at which the cement paste is described as a two-phase material composed of a solid cement phase and saturated capillary pores. The proposed multi-scale methodology uses an asymptotic homogenization framework, which is formulated within a multi-physics scheme to describe the coupled mechanical and diffusive responses. This approach is new for cementitious materials, and is characterized by two consecutive steps: the homogenization method is first applied with respect to a micro-structural unit cell, which provides the effective meso-scale elastic stiffness and diffusivity of the porous cement paste as a function of capillary porosity. Together with the properties of the aggregate, these properties are subsequently used as input for a meso-structural unit cell, which is subjected to a second homogenization step in order to calculate the effective elastic, diffusive and chemo-expansive macroscopic properties of concrete. The numerical analyses illustrate that the homogenization step between the microscopic and mesoscopic levels provides a realistic estimate of the effective elastic modulus and diffusion coefficient of the porous cement paste. The effective stiffness and diffusivity decrease and increase as a function of porosity, respectively, and show to be in good correspondence with experimental data taken from the literature. The simulations further reveal that the effective expansivity of the microstructure is independent of the porosity, and is solely prescribed by the expansivity of the solid cement phase. This is due to the fact that the pores do not constrain a volumetric change of the cement paste. The presence of percolation channels within the micro-structure, which in

practice is typically observed for cements with moderate to large porosities, may cause a drastic reduction of the effective elastic stiffness of the meso-scale unit cell, and a significant increase of its effective diffusion coefficient. The results of the homogenization step between the mesoscopic and the macroscopic levels illustrate an increase of the effective elastic modulus as a function of the aggregate volume fraction, which agrees well with experimental data taken from the literature. Both the effective macroscopic expansivity and macroscopic diffusivity decrease under an increasing aggregate volume fraction. The calculated effective diffusivity is in good correspondence with experimental data obtained from literature. Further, the influence of the stiffness mismatch between the aggregate and the cement on the effective macroscopic properties is substantial for values in between 10−2 and 102. For stiffness mismatches falling outside this range, the effective properties change only mildly, and approaches an asymptotic value. The effective mechanical, diffusive and chemo-expansive properties calculated with the current approach can be used as input parameters for macroscopic numerical analyses that aim at predicting the response of concrete structures subjected to complex chemical and mechanical loadings, as for instance occurs under chloride or sulphate attacks. As a topic of future research, the effective properties of concrete can be extracted from micro-structural and meso-structural models based on SEM images of real concrete and cement paste materials.

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Acknowledgements

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This work is part of the research programme “A NumericalExperimental Study on chemo-mechanical degradation of COncrete sewer Pipes (NESCOP)” with project number 15485, which is financed by the Dutch Research Council (NWO). References

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