A mixture theory-based concrete corrosion model coupling chemical reactions, diffusion and mechanics
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Abstract
A 3-D continuum mixture model describing the corrosion of concrete with sulfuric acid is built. Essentially, the chemical reaction transforms slaked lime (calcium hydroxide) and sulfuric acid into gypsum releasing water. The model incorporates the evolution of chemical reaction, diffusion of species within the porous material and mechanical deformations. This model is applied to a 1-D problem of a plate-layer between concrete and sewer air. The influx of slaked lime from the concrete and sulfuric acid from the sewer air sustains a gypsum creating chemical reaction (sulfatation or sulfate attack). The combination of the influx of matter and the chemical reaction causes a net growth in the thickness of the gypsum layer on top of the concrete base. The model allows for the determination of the plate layer thickness h=h(t) as function of time, which indicates both the amount of gypsum being created due to concrete corrosion and the amount of slaked lime and sulfuric acid in the material. The existence of a parameter regime for which the model yields a non-decreasing plate layer thickness h(t) is identified numerically. The robustness of the model with respect to changes in the model parameters is also investigated.
Keywords
Reaction-diffusion Mechanics Mixture theory Concrete corrosion Sulfatation attackMathematics Subject Classification (2010)
Primary 74D05 74F20 secondary 74B05 74F10 74F25PACS Codes
Primary 47.70.Fw 83.10.Ff secondary 46.70.-p 47.10.Ab 83.60.BcIntroduction
Forecasting concrete corrosion is a major issue in civil engineering due to its potential of drastically decreasing the lifespan of constructions such as sewers, bridges and dams, see e.g. [13, 33, 35]. As an example, the differences in mechanical properties between gypsum and concrete result in volume expansion, cracking, and decrease in load-bearing capacity of the concrete resulting in compromised structural integrity followed by expensive repairs, construction replacements or even accidents due to (partial) collapse [19, 38] resulting in major costs for society [14, 39].
We focus on three related topics: Firstly, we aim to construct a 3-D continuum mixture model describing concrete corrosion capable of exhibiting realistic behaviour of the growth of a concrete layer due to the formation of gypsum inside the concrete layer. Secondly, we apply the new model to a specific 1-D situation of the concrete layer and investigate the validity of the behavior of this 1-D model with respect to physical constraints and expected physical behavior. Finally, we investigate the parameter dependence of both the time span of realistic behavior and growth of the concrete layer for the 1-D model.
Even though concrete is a heterogeneous material, a lot of research has been done relying on continuum models, where the heterogeneity details are averaged out. In [26] the reader can find a short historical overview of the use of continuum models in concrete research. Similar to the continuum models from [26], the authors of ref. [31] proposed a composite material model of concrete with an explicit volume division into mortar and aggregate. These models were mostly created to better describe the behavior of concrete under high stresses, and, hence, to predict the cracking behavior observed in the experiments reported in [26, 31]. The mathematical community has addressed this corrosion issue mainly from a single-scale or multiple-scale reaction-diffusion perspective. Usually, the single scale approach involves one or two moving sharp reaction interfaces [1, 12, 17, 18, 29, 30], while the multiple scale setting prefers exploiting a better understanding of the porosity and tortuosity of the material without involving free boundaries [2, 10, 16]. There are still a number of open issues concerning on how poro-mechanics of the material couples with chemical reactions, flow, diffusion and heat transfer hindering a successful forecast of the durability of the concrete exposed to sulfate attack. In this paper, we are interested in understanding and then predicting eventual critical situations occurring before cracking. Particularly, we want to describe the corrosion of concrete by acid attack [37], which usually leads at a later stage to cracking followed by erosion. The main inspiration source for our problem setting is the basic scenario described in [4] which considers a simple reaction mechanism producing gypsum, without involving the ettringite formation.
In [4] an isothermal acid attack continuum model for sulfuric acid corrosion was proposed with a similar sewer pipe geometry as in our model, but including also the porosity of the gypsum. This model focussed solely on the creation of hydrogen sulfide and sulfuric acid, which reacts at the boundary to create gypsum. The model assumed that almost all the gypsum was created at the boundary separating the uncorroded concrete causing a moving sharp corrosion front penetrating irreversibly the material. We deviate from this model by assuming that the gypsum reaction gradually takes place in the full domain, and that the corrosion front is caused by the penetration of sulfuric acid. In some sense our model can be seen as a description of the moving corrosion front in [4] as a fixed bulk reaction domain, and can, therefore, be idealized into a plate-layer model. To avoid describing the exact growth of the involved phases of the material, we take a modeling route in the spirit of the classical mixture theory.
It is worth noting that most of the assumptions mentioned in [4] are taken over here as well. For example, our model is supposed to reflect the entire corrosion process with no other contributing chemical reactions and species than those explicitly mentioned. Also, the external concentration and influx rates of sulfuric acid and hydrogen sulfide are constant. Both these assumptions are restrictive. For example, competing corrosion reactions and other reacting chemicals, such as nitrates, are present in an actual concrete corrosion process according to [4]. Moreover, in [9] it is explained that experiments show that external concentrations and influx rates are not even approximately constant because flow changes (changing Reynolds number) have enormous influences, which according to [4] could change rates and concentrations with many powers of 10. Hence, the assumptions of ref. [4] are necessary to reduce the complexity of our model.
Our paper is organized as follows. In Section 2, we construct several 3-D continuum mixture models of chemical corrosion of concrete. We take into account effective balance laws, diffusion processes, chemical reaction effects, mechanical effects due to elastic and/or viscoelastic stresses, local interactions due to for instance the Stokes drag, and influx from external reservoirs and from domain growth due to a moving corrosion layer. In Section 3, we focus on the normal (z) direction to obtain an effective 1-D model of the corroding concrete for one of the constructed models. In Section 4, we briefly describe both the code used to simulate the model of Section 3 and the validation of this code with respect to the asymptotic expansion solution obtained in Appendix A. In Section 5, we investigate the validity of the numerical behavior of the model of Section 3. In Subsection 5.1, we investigate the dependence of the realistic behavior on specific tuples of model parameters. Finally, in the conclusion we summarize our results and discuss the relation of these results with known literature.
Derivation of a mixture-theory-based concrete corrosion model
The presentation of a continuous 3-component mixture model in this section is based on the theory of mixtures of Bowen in [7].
2.1 Preliminaries
We assume that the constituents of the mixture are incompressible. Hence, the intrinsic densities \(\tilde {\rho }_{\alpha }\) are uniform constants.
2.2 Balance laws
Following [7] and in analogy with [6, 27], we describe the time evolution of our 3-component mixture by means of two sets of global balance laws for each component of the mixture: one for mass and one for momentum conservation. We assume that the chemical reaction is an isothermal process; the conservation of energy is then automatically satisfied.
Note, the elements g(t) can overlap due to the diffusion term, but that it does not violate the material element rules.
Since a chemical reaction is inherently a mass-conserving process, we obtain \(\sum _{\alpha } R_{\alpha } = 0\). Thus this global mass conservation is satisfied if \(\sum _{\alpha }\delta _{\alpha }\nabla \rho _{\alpha } = \sum _{\alpha }\delta _{\alpha }\tilde {\rho }_{\alpha }\nabla \phi _{\alpha }=0\), a compatibility condition for the allowed types of internal diffusion processes. This is satisfied if, for instance, \(\delta _{\alpha } = \delta /\tilde {\rho }_{\alpha }\). Hence, δ=0 (no internal diffusion) would suffice.
where ρ_{α}v_{α} is the linear momentum density of the component α, while the outward flux is given by the partial stress tensor \(\mathbb {T}_{\alpha }\) and the production term by the internal linear momentum production B_{α}. The latter two terms will be specified in the next subsection. Since in our setting the mechanical processes and flow dynamics are slow, we assume a quasi-static regime. This implies that the inertia term on the left-hand side in Eq. 8 may be neglected. Moreover, the sum of the internal momentum-production terms B_{α} must be zero, i.e. \(\sum _{\alpha }\mathbf {B}_{\alpha }=0\), by Newton’s third law.
2.3 Local equations and jump conditions
with use of \(\delta _{\alpha } = \delta /\tilde {\rho }_{\alpha }\). We refer to Eq. 12 as the incompressibility condition. Later we shall use Eq. 12 to replace one of the three mass equations (e.g. for α=2, and then use ϕ_{2}=1−ϕ_{1}−ϕ_{3}).
Before we can evaluate the local momentum equations any further we have to make constitutive assumptions concerning the structure of \(\mathbb {T}_{\alpha }\) and B_{α}.
where \(\mathbb {D}_{\alpha } = \left (\nabla \mathbf {v}_{\alpha }+(\nabla \mathbf {v}_{\alpha })^{\top }\right)/2\) is the rate of deformation tensor based on the velocity v_{α}=∂u_{α}/∂t, while the coefficients γ_{αβ} are material constants that will be further specified below.
Table with numerical values of material constants, normalization constants, dimensionless parameters, and numerical parameters
Material constants | Dimensionless parameters | |||||
---|---|---|---|---|---|---|
Value | (MKS unit) | Reference | Value | Definition | ||
E _{1} | 1.60 ·10^{9} | (kg/m s^{2}) | [32] | E _{1} | 0.038 | E_{1}/E |
E _{2} | 4.20 ·10^{10} | (kg/m s^{2}) | [41] | E _{2} | 1.00 | E_{2}/E |
χ _{1} | 2.67 ·10^{10} | (kg/m^{3}s) | † | χ _{1} | 1.00 | χ_{1}/χ |
χ _{2} | 2.67 ·10^{10} | (kg/m^{3}s) | † | χ _{2} | 1.00 | χ_{2}/χ |
\(\mathcal {J}_{2}\) | 0.326 ·10^{−5} | (m/s) | * | \(\mathcal {J}_{2}\) | 0.40 | \(\mathcal {J}_{2}/J\) |
\(\mathcal {J}_{3}\) | 1.632 ·10^{−5} | (m/s) | * | \(\mathcal {J}_{3}\) | 2.00 | \(\mathcal {J}_{3}/J\) |
γ _{1} | 3.604 ·10^{10} | (kg/ms) | * | γ _{1} | 0.50 | γ_{1}/γ |
γ _{2} | 3.604 ·10^{10} | (kg/ms) | * | γ _{2} | 0.50 | γ_{2}/γ |
A _{1} | 0.821 ·10^{−3} | (1/m) | * | A _{1} | 0.50 | A_{1}/A |
A _{2} | 0.821 ·10^{−3} | (1/m) | * | A _{2} | 0.50 | A_{2}/A |
\(\tilde {\rho }_{1}\) | 2.32 ·10^{3} | (kg/m^{3}) | [20] | ϕ _{1 s a t} | 1.00 | |
\(\tilde {\rho }_{2}\) | 2.21 ·10^{3} | (kg/m^{3}) | [20] | ϕ _{3 t h r} | 0.00 | |
\(\tilde {\rho }_{3}\) | 1.84 ·10^{3} | (kg/m^{3}) | [20] | ϕ _{2 r e s} | 1.00 | |
\(\mathcal {M}_{1}\) | 0.172164 | (kg/mol) | [20] | ϕ _{3 r e s} | 1.00 | |
\(\mathcal {M}_{2}\) | 0.074093 | (kg/mol) | [20] | κ _{1} | 23.00 | Eq. 51 |
\(\mathcal {M}_{3}\) | 0.098079 | (kg/mol) | [20] | κ _{3} | 13.50 | Eq. 51 |
δ | 5.10 | (kg/m s) | * | δ _{1} | 1.00 | δ_{1}/KH^{2} |
δ _{1} | 2.20 ·10^{−3} | (m^{2}/s) | ‡ | δ _{2} | 1.05 | δ_{2}/KH^{2} |
δ _{2} | 2.31 ·10^{−3} | (m^{2}/s) | ‡ | δ _{3} | 1.26 | δ_{3}/KH^{2} |
δ _{3} | 2.77 ·10^{−3} | (m^{2}/s) | ‡ | |||
k | 1.00 ·10^{−6} | (m^{3}/mol s) | [3] | |||
Normalization Constants | Numerical Parameters | |||||
Value | (MKS unit) | Definition | Value | Definition | ||
H | 1.643 ·10^{0} | (m) | h(0) | Δ t | 0.001 | |
K | 0.816 ·10^{−3} | (1/s) | Eq. 44 | t _{ f} | 0.5 | T_{f}/T |
S _{ K} | -1 | (-) | Eq. 44 | 1/Δz | 300 | |
χ | 2.67 ·10^{10} | (kg/m^{3}s) | χ _{1} | ϕ _{min} | 10^{−5} | |
E | 4.20 ·10^{10} | (kg/m s^{2}) | E _{2} | V _{max} | 10^{6} | |
T | 1.716 | (s) | χH^{2}/E | |||
U | 2.300 ·10^{−3} | (m) | χH^{3}K/E | |||
V | 1.341 ·10^{−3} | (m/s) | HK | |||
J | 0.816 ·10^{−3} | (m/s) | HK | |||
γ | 7.208 ·10^{10} | (kg/m s) | χ H ^{2} | |||
ε | 0.0014 | (-) | χH^{2}K/E |
- 1System A: This system corresponds best to the evolution systems studied in [40], where conditions for the existence of weak solutions were obtained. Here, the individual constituents are assumed to be viscoelastic, such that the mixture as a whole remains purely elastic. For this, we choose γ_{αβ}=γ_{α} if β=α∈{1,2}, and γ_{αβ}=0 if β≠α, resulting in$$ \mathbb{T}_{\alpha}^{\text{ve}}=\gamma_{\alpha}\mathbb{D}_{\alpha}\;\text{ for }\alpha \in \{1,2\}. $$(22)Moreover, we take \(\mathbb {T}_{3}^{\text {ve}}\) such that$$ \mathbb{T}_{3}^{\text{ve}}=-\sum_{\alpha=1}^{2}\gamma_{\alpha}\mathbb{D}_{\alpha}=-\sum_{\alpha=1}^{2}\mathbb{T}_{\alpha}^{\text{ve}}, $$(23)
providing that \(\mathbb {T} = \sum _{\alpha =1}^{3}\mathbb {T}_{\alpha }^{\text {ve}}=0\).
- 2
System B: Here, γ_{αβ}=0: the solid components are thus purely elastic and the fluid inviscid.
- 3
System C: As in System A, the solid components are intrinsic viscoelastic, but the fluid is inviscid, so \(\mathbb {T}_{3}=-\phi _{3}p\mathbb {I}\), implying that the mixture as a whole is also viscoelastic. This has consequences on the pressure term p, as can be seen in the 1-D problem described in Section 3; see (41).
- 4System D: In this case, we assume that the viscoelastic terms in the stresses are proportional to the differences in shear rates of the two solids so that these stresses are zero if the velocities, or displacements, of the solids are equal. Moreover, we let the sum of the two stresses equal zero and keep the fluid inviscid. Thus, the total stress is purely elastic. This results in the following choice for γ_{αβ}$$ \gamma_{11}=\gamma_{22}=\gamma,~~~\text{and}~~~\gamma_{12}=\gamma_{21}=-\gamma. $$(24)
System A is well-posed mathematically (cf. [40]), but is possibly physically incorrect as the sulfuric acid viscoelastic stress is defined by the viscoelastic stress of the other components, see (23). System B is physically nice, but mathematically it needs an additional viscoelastic term to ensure the existence of weak solutions and FEM approximations. System C combines the strong points of systems A and B. It is physically justified and mathematically sound. However, the mixture is viscoelastic, which is a behavior one would expect on unnaturally large timescales. System D is both mathematically and physically sound, supporting an elastic mixture, which favors timescales compatible with measurements.
The physical derivation of systems A, B, C and D indicate that only system D has the right physical properties at the desired timescales. Hence, from here on we will focus on system D from both analytical and numerical perspectives, for example when we judge solutions to exhibit realistic behaviors. To reduce complexity, we investigate a special situation leading effectively to a 1-D version of system D.
2.4 Chemical corrosion of concrete with sulphates
Hence, the stoichiometric coefficients N_{α} are N_{1}=1 and N_{2}=N_{3}=−1.
where we denote \(\mathcal {L}(u) = u\mathcal {H}(u)\) with \(\mathcal {H}\) the Heaviside function, k is the volumetric reaction rate (in [m^{3}/mol ·s]), [f] the molar concentration of f, C_{eq} the dissolution equilibrium molar concentration of the sulfuric acid, and C_{max} the maximum precipitation molar concentration of gypsum.
where ϕ_{1,sat} is the gypsum saturation level, while ϕ_{3,thr} represents the sulfuric acid dissolution threshold.
2.5 Initial and boundary conditions
where \(\phi _{\alpha 0} = \rho _{\alpha 0}/\tilde {\rho }_{\alpha }\) are prescribed initial concentration values.
We wish to point out here that, although u_{3}(x,0^{+})=0, there is a jump in the velocity v_{3}, which is inherent to the quasi-static approximation we used.
the right-hand side of which is greater than zero if \(\phi _{\beta }^{+}>\phi _{\beta }\).
which is equivalent to requiring that the partial normal stress of constituent α is zero.
2.6 Summary of the model equations
in which the γ-term is only non-zero for System C.
We can replace Eq. 37d describing the fluid motion by this global equation, and then determine the pressure p from it with the aid of the stress boundary condition.
The initial conditions are given in Eq. 29 and the necessary boundary conditions are Eqs. 30, 31, 33 and 35.
Dimension reduction: 1-D model of a concrete plate-layer
We reduce the 3-D model of Section 2 to a simpler 1-D problem, namely a flat plate-layer of concrete of initial thickness H, which is exposed at its upper side to acidic air due to the presence of droplets of sulfuric acid. The bottom of the plate layer is fixed on a rigid ground space of non-reacting concrete having a fixed concentration of lime. The material of the layer (concrete) is a mixture of gypsum (α=1), lime (α=2) and sulfuric acid (α=3). Initially, i.e. for t<0, the layer is in a homogeneous, undeformed, and stress-free state, where the sulfuric acid has penetrated the concrete and has already partially reacted to create gypsum, such that ϕ_{α0}>0 for α=(1,2,3). The external space both below and above the plate is free of stress. As the layer is created in a homogeneous and uniform way, and the acid is in equilibrium, we can forget about the tangential directions and only focus on the normal (z) direction. Hence, a 1-D plate-layer model is sufficient to model a 3-D sewer pipe as already explained in the Introduction.
instead of γ. Since these effective coefficients depend on the volume fractions ϕ_{1,2} the (numerical) analysis of this system becomes more complicated than for the other systems.
as they follow from (31), (32) and (35), respectively. We notice that we need in total 9 boundary conditions (2 for each of ϕ_{1}, ϕ_{2}, u_{1}, u_{2} and 1 for v_{3}), as well as an extra condition to determine h(t), so in total 10 conditions.
3.1 Dimensionless formulation
From both these results we conclude that the first two terms, the influxes with \(\mathcal {J}_{2,3}\) being positive, yield a positive contribution to \(\mathcal {W}(t)\) making the layer increase in thickness. Whether or not the third term has an increasing or decreasing effect depends on the sign of S_{K}; when, as in our case, S_{K}=−1, the chemical reaction does shrink the layer. At this moment, nothing specific can be said for the last term. However, our numerical results reveal that the effect of this term is always small. Thus, we can state that the domain of the layer only grows if the magnitude of the first two terms is greater than the third one. Hence, there is a competition effect here.
In Appendix A, a solution for System D has been obtained as a formal asymptotic expansion in ε. The asymptotic expansion is formal as it is not a priori known whether or not this expansion is converging in ε. The predictive power of a formal asymptotic expansion should not be underestimated, because there exist formal asymptotic expansions, which are diverging, but can be very accurate when only a truncated version of the expansion is used; see the example in Section 1.4.2 on pages 13 and 14 of [21]. This motivated us in the choice of the two J-parameters; see Table 1.
Numerical method
In this section, we solve numerically the systems A, C and D. We omit system B, because a viscoelastic term is needed to obtain a coercive system, such as in system A, for which we have proven the convergence of the time-discrete evolutions to the corresponding weak solution; see [40]. We expect that similar convergence results can be obtained for the systems C and D, as they have a viscoelastic term similar to the one in system A. Also, when solving system D we exclude the Laplacian terms in Eq. 43c, or stated in another way: the numerical method uses δ_{α}=0 for (43c). This exclusion is justified by an order analysis of the terms of (43c) from the ϕ_{α}-solutions of (43a) and (43b), which states that \(\sum _{\alpha =1}^{3}\delta _{\alpha }\Delta \phi _{\alpha } = \mathcal {O}\left (\sqrt {\epsilon }\mathcal {F}\right)\).
Our code is called NewGypsum and it is based on a combination of MATLAB routines. We start off with a Rothe time discretization of the systems A, C and D, which linearizes the systems. Benefitting from the one-dimensional-in-space formulation, solving the linear systems is done automatically by using the built-in boundary value problem (BVP) solvers of MATLAB, see bvp4c and bvp5c; [22, 23]. These solvers take a grid, a guess for the solution, and the BVP system as input. Then they automatically readjust the grid and interpolate the guess solution to obtain a starting point for the numerical scheme, controlling a certain error metric to determine the solution based on user-defined-convergence criteria.
The solver bvp4c is an implicit Runge-Kutta method using the 3-stage Lobatto IIIa formula with control on the residual [22]. The method is only applicable to linear Lipschitz systems [22]. Fortunately, systems A, C, and D can be shown to satisfy this condition within certain parameter constraints (which we will explain more thoroughly in the next section). For an easy guide in understanding and using bvp4c we recommend [36]. Moreover, [36] shows that boundary layer effects are well resolved by the bvp4c solver.
The solver bvp5c is an implicit Runge-Kutta method using the 4-stage Lobatto IIIa formula with control on the true error [23]. The solver bvp5c is more precise than bvp4c, but it is also less versatile [23]. This does not pose a problem as our three systems A, C and D still satisfy the applicability conditions for bvp5c and bvp5c has similar capabilities in handling boundary layers as bvp4c [23]. In our case the choice was made to use bvp5c as it made our simulations about 27 times faster than when using bvp4c.
A more detailed explanation of our NewGypsum can be found in Section 2.4 of [40]. Moreover, in Appendix A one can find a validation of the NewGypsum routine with a Mathematica simulation of the asymptotic ε-expansion solutions derived in the same appendix.
Quest for realistic numerical behavior
- 1The volume fractions should be nonnegative and less than one. From the mathematical analysis point of view we expect that system A behaves poorly when volume fractions become very small. To outlaw this unwanted behavior a positive minimal value ϕ_{min} is introduced, leading to the constraint$$ 0<\phi_{\min}\leq\phi_{\alpha}(t,z)<1 $$(56)
for all α∈{1,2,3}, for all z∈(0,1), and for all t∈(0,t_{f}).
- 2A second condition is a demand on the upper bound for the velocity. Fast local deformations are allowed as long as the total contribution to the domain deformation is still small, the stresses remain low and the quasi-static approximation is not violated. Hence, it is natural to cap both the total velocity in the domain and the total spatial change of the velocity in the domain. This is reflected in the condition$$ {\begin{aligned} \|v_{3}\|_{L^{2}(t_{0},t;H^{1}(0,1))}^{2} = \int_{0}^{t}\left[ \int_{0}^{1}\left(v_{3}(s,z)^{2}+\left(\partial_{z}v_{3}(s,z)\right)^{2}\right)\text{d}z\right]\text{d}s<V^{2} \end{aligned}} $$(57)
for all t∈(0,t_{f}).
- 3The concrete layer has two boundaries that allow influx. Even though the chemical reaction itself is volume contractive, the combination of influx and chemical reactions must be volume expansive due to the porous nature of gypsum [28]. Hence, the height of the plate-layer must be a nondecreasing function:$$ \partial_{t}h(t) = \epsilon\partial_{t}\mathcal{W}(t)\geq0\quad\text{for all }t\in(0,t_{f}). $$(58)
Realistic behavior is defined as satisfying all three constraints Eqs. 56 to 58. We immediately stop a simulation when one of the three inequalities is violated.
We need a benchmark of our numerical program to test the numerical solutions for realistic behavior. For this we introduce a reference set of material constants. The values of these constants, and their dimensionless counterparts, dimensionalized with respect to the diffusion time scale, are listed in Table 1. The numerical evaluations use a time step Δt, the size of the time interval t_{f}, and a number of spatial subdivisions, 1/Δz. We choose fixed values Δt=0.001, t_{f}=0.500 and 1/Δz=300 for these parameters. In the remainder of this paper we implicitly use the parameter values of Table 1, whenever parameter values are not explicitly specified. A spatial-temporal analysis of our benchmark problem with the parameter values of Table 1 can be found in Section 2.6 of [40], showing that our benchmark simulation gives expected behaviour.
5.1 Parameter dependence of found realistic behavior
We aim to determine how the size of the realistic time interval, given in number of numerical iterations N_{R}, depends on the system parameters. Our definition of realistic behavior contains three constraints, see the beginning of Section 5, which can be numerically checked. We investigate the numerical simulation applied to systems A, C and D for a large parameter range, by changing specific parameters in Table 1. In this way our results even hold when experimental values with large uncertainties are used for the model parameters if these values with uncertainties remain in the probed region. Out of the 20 model parameters, we will only change specific parameters chosen on basis of their influence on the analytical bounds in the existence proof in [40]. When a bound in this existence proof contains a product of two parameters, then this parameter pair is chosen. All parameters are modified in a double exponential fashion such that large parameter ranges are investigated. Finally, the initial condition (ϕ_{10},ϕ_{20},ϕ_{30}) is chosen, because they immediately determine whether chemical reactions or influx do occur.
The parameter pair (A_{2},γ_{2}) should be investigated as well. However, we chose to fix the ratios A_{1}/A_{2} and γ_{1}/γ_{2}, because the dependence on (A_{2},γ_{2}) is expected to be similar to the dependence on (A_{1},γ_{1}). Similarly, we chose to fix the ratios \(\mathcal {J}_{2}/\mathcal {J}_{3}\) and ϕ_{2,res}/ϕ_{3,res}. Moreover, if parameters are not mentioned to have special values, then these parameters are set to their standard values as listed in Table 1.
The existence proof in [40] points out a dependence on the (κ_{1},κ_{3},ϕ_{1,sat}) parameter triple. However, the dependence on ϕ_{1sat}, κ_{1} and κ_{3} is quite subtle: only for ϕ_{1,sat}>ϕ_{1}≈ϕ_{10} the chemical reaction is active and \(\mathcal {F}>0\). This has only a relevant effect on the incompressibility condition, because in the first two diffusion equations Eqs. 50a and 50b the right-hand sides are of \(\mathcal {O}(\epsilon)\). This implies that the effect of κ_{1} and κ_{3} on the simulations is expected to be (negligibly) small. As we made not enough simulations for ϕ_{1} above the ϕ_{1,sat} threshold value, we can not draw any conclusions concerning its effect on realistic behavior. However, we expect an increasing ϕ_{1,sat} to decrease the size of the realistic time interval, as increasing ϕ_{1,sat} increases the size of \(\mathcal {F}\) and, hence, also the size of v_{3}.
The three systems behave differently as one can see from the size of the parameter region with 500 iterations. The parametric region pointing at the high acid concentration region is outperforming the other parameter regions in all systems. A high concentration of acid implies that the reaction is slow (i.e. \(\mathcal {F}\) is small), and consequently, the velocity v_{3} remains small. Moreover, also the influx of acid is low or even absent. This results in a relatively small increase of the norm of v_{3}, and, therefore, violating the velocity norm upper bound (which is the most critical of the three conditions to violate) takes more time for large values of ϕ_{3}. This explains the good performance of this parameter region.
For the determination of the dependence on other parameters the best choice of initial conditions for each system is exactly in the transition region between the regions of small (single digit) and high (500) amount of iterations. In this transition region, the amount of iterations is expected about half way in between 1 and 500 iterations. Any dependence yielding lower or higher amounts of iterations is faithfully represented. Outside this transition region the registration of the dependence is limited to a one-sided deviation of the reference level of amount of iterations, while in this transition region the registration allows for the full two-sided deviation of the reference level of the amount of iterations. We have chosen (ϕ_{10},ϕ_{20},ϕ_{30}) equal to (11/30,11/30,8/30), (1/3,1/3,1/3), and (1/4,1/4,1/2) for System A, C, and D, respectively.
Number of consecutive iterations yielding realistic behavior for Systems A, C and D at different values of δ
δ=1.00 × factor below | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
System | 10^{−5} | 10^{−4} | 10^{−3} | 10^{−2} | 10^{−1} | 1 | 10 | 10^{2} | 10^{3} | 10^{4} | 10^{5} |
A | 297 | 304 | 297 | 311 | 311 | 324 | 332 | 338 | 331 | 338 | 338 |
C | 212 | 222 | 220 | 216 | 218 | 220 | 216 | 212 | 230 | 222 | 212 |
D | 462 | 462 | 462 | 462 | 464 | 464 | 464 | 464 | 464 | 464 | 464 |
For all systems, we see that the size of δ has practically no influence and is, therefore, unimportant in establishing realistic behavior defined in this section. This makes sense because the initial conditions are smooth, which leads to small values of the Laplacian. Hence, δ has only a minor effect on the simulation output.
Unbiased estimators of α_{0} and their standard error for the relationship \(\protect \phantom {\dot {i}\!}T_{real} \sim \epsilon ^{\alpha _{0}}\) describing the dependence of the realistic time interval of System D on the parameter ε for two different initial conditions
System D: (ϕ_{10},ϕ_{30}) | (0.20,0.50) | (0.25,0.50) |
---|---|---|
\(\hat {\alpha }_{0}\) | −0.509 | -0.487 |
\(s_{\hat {\alpha }_{0}}\) | 0.00854 | 0.0121 |
# datapoints | 7 | 7 |
The realistic behavior is affected by changes in \(\mathcal {J}_{\alpha }\), α∈{2,3}, as they control the rate of influx and so a major aspect of thickness growth. Increasing the size of \(\mathcal {J}_{\alpha }\) gives a corresponding increase in the size of \(\mathcal {W}(t)\) for large enough \(\mathcal {J}_{\alpha }\). However, for small \(\mathcal {J}_{\alpha }\) we cannot expect the same correspondence, because at some point the reaction becomes the dominant contributor. Hence, for small \(\mathcal {J}_{\alpha }\) the growth of \(\mathcal {W}(t)\) must be independent of \(\mathcal {J}_{\alpha }\), while at large \(\mathcal {J}_{\alpha }\) this growth must be in a one-to-one correspondence.
Number of consecutive iterations yielding realistic behavior (N_{R}) for Systems A, C and D, and a set of values for the parameter pair (A_{1},γ_{1})
System A | System C | System D | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
2γ_{1} | 2γ_{1} | 2γ_{1} | ||||||||||
2A_{1} | 0.01 | 0.1 | 1 | 10 | 0.01 | 0.1 | 1 | 10 | 0.01 | 0.1 | 1 | 10 |
0.1^{5} | 12 | 13 | 57 | 500 | 16 | 17 | 33 | 500 | 410 | 410 | 410 | 412 |
0.1^{4} | 12 | 13 | 57 | 500 | 16 | 17 | 33 | 500 | 410 | 410 | 410 | 412 |
0.1^{3} | 12 | 13 | 59 | 500 | 16 | 17 | 33 | 500 | 410 | 410 | 410 | 412 |
0.1^{2} | 12 | 13 | 59 | 500 | 14 | 17 | 33 | 500 | 410 | 410 | 412 | 412 |
0.1 | 10 | 14 | 123 | 500 | 14 | 19 | 38 | 500 | 416 | 416 | 416 | 418 |
1 | 8 | 10 | 324 | 1 | 12 | 14 | 220 | 45 | 462 | 464 | 464 | 464 |
10 | 8 | 1 | 4 | 6 | 10 | 1 | 4 | 8 | 244 | 320 | 308 | 306 |
10^{2} | 2 | 6 | 8 | 8 | 2 | 4 | 8 | 8 | 1 | 1 | 1 | 1 |
The realistic behavior depends also on A_{α}. When A_{α} takes large values, then the coupling between \(\mathcal {W}(t)\) and the displacements u_{1} and u_{2} becomes strong, leading to a larger value of v_{3}, and thus smaller N_{R}. On the other hand, when A_{α} is small (say A_{α}<1), then the boundary condition will behave more like a Neumann boundary condition, having no effect whatsoever on the realistic time interval. Again, we see these behaviors in Table 4 for Systems A, C, and D. This behavior agrees with the analytical results from [40] for System A.
Conclusion
We have derived, based on first principles, several models describing concrete corrosion by taking into account mixture theory, small deformations, compressibility and viscoelastic effects, diffusion, chemical reactions, influx of chemical species and an expanding domain. The most suitable model is System D. For this system, we could obtain the best numerical results with nice power law behaviors, which lead us to the hypothesis that the realistic time interval T_{real} scales as \(1/\sqrt {\epsilon }\). Moreover, we could interpret the spatial behavior of all variables by taking into account the physical effects of the chemical reaction and of the influx of reacting materials.
- (i)
By performing more specific measurements at the length scale of our domain;
- (ii)
By upscaling procedures, obtain effective material coefficients at length scales compatible with the measurements;
- (iii)
By suitably combining (i) and (ii).
By performing simulations with intentionally large parameter ranges, we localized the uncertainties in the model parameters and probed simultaneously the continuous dependence of the solution to our systems on the choice of parameters. In this way, the behavior of System D is valid, even for the model parameters with large uncertainties. While probing the parameter dependence of our system on 20 different parameters, of which about 10 are indeterminate, we immediately encounter the curse of dimensionality – sampling a high dimensional space^{2} is a sparse operation. A more structured sampling was possible by targeting the variables present in analytical upper bounds derived in [40]. An additional complication is the nonlinear coupling of all unknowns involved concurrently in several physical processes. Such a strong coupling prohibits a fast simulation at a single parameter tuple and creates a complex nonlinear parameter dependence of the solution behavior.
What concerns System D, at least for a short transient time the realistic behavior showed practically constant concentrations due to the slow reaction with respect to the influx. The displacements and velocities seemed consistent with the influx of material, while the thickness of the concrete layer was growing steadily, as expected from real world observations. Moreover, these results coincide with [4] as the plate thickness increases in time and the correct changes in volume fractions were observed. Displacements and velocities could not be related to any quantity in [4], because their reaction occurs in the boundary, while ours occurs in the full domain.
The Systems A, C, and D showed strong dependence on several parameters. For all systems the number of consecutive iterations yielding realistic behavior (N_{R}) is highly dependent on the choice of ϕ_{30}, due to the incompressibility condition, while ϕ_{10} and ϕ_{20} seem unimportant, as long as ϕ_{10}+ϕ_{20}=1−ϕ_{30}. The diffusion coefficient δ>0 had no effect on N_{R}, while the scale separation parameter ε greatly influenced N_{R} for all systems, especially for System D with an apparent power law dependence. The reaction parameters κ_{1}, κ_{3}, ϕ_{1,sat} had no influence on N_{R}, because ε is small and \(\mathcal {J}_{3}>1\). The flux parameters \(\mathcal {J}_{2}\) and \(\mathcal {J}_{3}\) are unimportant at small values (\(\mathcal {J}_{2}<1\)), while almost in one to one correspondence with N_{R} at large values (\(\mathcal {J}_{2}>10\)) due to Eq. 54. The external concentrations ϕ_{res} had almost no influence on N_{R}, what can be attributed to an under sampling of large values (ϕ_{res}>0.3). The viscoelastic parameters γ_{1} and γ_{2} are important for keeping coercivity. They show a high dependence on N_{R} for Systems A and C, but almost no dependence for System D. The boundary condition parameters A_{1} and A_{2} highly influence N_{R}, but for Systems A and C the behavior seems erratic, except at small values due to the convergence to Neumann boundary conditions. The thickness \(\mathcal {W}(t)\) for System D becomes larger for smaller values of ε, but changes behavior for ε<0.0014, for which \(\mathcal {W}(t)\) seems independent of ε. This behavioral change is unexpected and advocates for additional research. Moreover, the thickness \(\mathcal {W}(t)\) increases continuously as expected from experiments.
Hence, the important parameters of Systems A, C, and D describing the behavior of N_{R} are ϕ_{30}, ε, \(\mathcal {J}_{\alpha }\), γ_{β} and A_{β} for α∈{2,3} and β∈{1,2}. Moreover, the observed behavior of the thickness \(\mathcal {W}(t)\) is largely as expected from observations.
Appendix A
7.1 Asymptotic ε-small solutions to System D
and the same for u_{α}(z,t;ε) and v_{3}(z,t;ε).
with \(\phi _{30} J_{3}:= \mathcal {J}_{3} \mathcal {L}(\phi _{3,res}-\phi _{30})\).
with b_{1}=−2b_{2}−3b_{3}, \(2\mathbf {b}_{2} = \hat {\mathbb {B}}^{-1}(\mathbf {J}-\hat {\mathbf {r}}_{1})\) and \(6\mathbf {b}_{3}= - \hat {\mathbb {B}}^{-1}\hat {\mathbf {r}}_{0}\), where \(\hat {\mathbf {r}}(z)=\mathbb {A}^{-1} \mathbf {r}(z)=:\hat {\mathbf {r}}_{1} + \hat {\mathbf {r}}_{0} z\). Note, \(\mathbb {A}\) and \(\mathbb {B}\) are invertible because χ_{1}χ_{2}ϕ_{30}≠0 and E_{1}E_{2}ϕ_{30}≠0, respectively.
Finally, we find v_{3}(z,t) from (65) and \(\mathcal {W}(t)\) from (69).
Footnotes
- 1.
In principle the right-hand side of Eq. 31 should be −ϕ_{α}(v_{α}−V) instead of 0. However, in our linear theory the value 0 is justified due to the scale separation between displacement and the actual size of the domain. See Section 3 for the effect of scale separation on the system in the dimension reduction process.
- 2.
In our case, the dimensionality is linked to the space of simulations for all possible combinations of parameter values.
Notes
Acknowledgements
We acknowledge NWO and NDNS+ for the funding leading to this manuscript.
Funding
This work was funded by the Netherlands Organization for Scientific Research (NWO) under contract no. NWO-MPE 657.000.004. Moreover, we acknowledge the NWO cluster Nonlinear Dynamics in Natural Systems (NDNS+) for funding a research stay of AJV at Karlstads University to visit AM. These funding bodies had no roles in the design of the study and collection, analysis, and interpretation of data and in writing.
Availability of data and materials
The simulation codes and simulation data will become available at an open repository, when this paper is accepted.
Authors’ contributions
The contributions of the authors are as followed: AJV 50%, AM 10%, FvdV 40%. All authors read and approved the final manuscript.
Ethics approval and consent to participate
The authors declare that the research presented in this manuscript complies with the ethics guidelines of their respective institutions. The research presented in this manuscript is devoid of participation by individuals.
Consent for publication
The authors declare that the research presented in this manuscript is devoid of personal data of individuals.
Competing interests
The authors declare that they have no competing interests.
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