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Modeling of Degradation Processes in Historical Mortars

The aim of presented paper is modeling of degradation processes in historical mortars exposed to moisture impact during freezing. Internal damage caused by ice crystallization in pores is one of the most important factors limiting the service life of historical structures. Coupling the transport processes with the mechanical part will allow us to address the impact of moisture on the durability, strength and stiffness of mortars. This should be accomplished with the help of a complex thermo-hygro-mechanical model representing one of the prime objectives of this work. The proposed formulation is based on the extension of the classical poroelasticity models with the damage mechanics. An example of two-dimensional moisture transport in the environment with temperature below freezing point is presented to support the theoretical derivations.

hygro-mechanical model representing one of the prime objectives of this work. The proposed formulation is based on the extension of the classical poroelasticity models with the damage mechanics. An example of two-dimensional moisture transport in the environment with temperature below freezing point is presented to support the theoretical derivations.

Introduction
Understanding the hydro-thermo-mechanical behavior of building materials exposed to weather conditions is the first step toward avoiding deterioration of structures in general and historical ones in particular, as high moisture content in building material and its phase changes are often a cause of internal damage.
Because of variable climatic conditions, the moisture gradients induce mechanical stresses in the porous material. These stresses mostly develop due to the growth of ice crystals through the pore structure. Therefore, there is a strong need for investigating the influence of moisture on the mechanical material behavior, which leads to numerical and experimental coupling of mechanical and thermo-hygro phenomena.
In the literature, the above described problem is addressed from several perspectives. The first group of publications is focused on the description of the coupled heat and moisture transport reflecting the moisture migration under the conditions of the ice crystal formation in the pores, 2-D and 3-D aspects and different moisture/heat sources, such as wind driven rain, solar short and long wave radiation etc., see [8,10,25]. An extensive overview of various transport models is available in [12,26]. While models for transport processes have been developed during several decades, the theory of ice crystallization in the pores has emerged only recently, [19,20,22]. The authors established relations between physical state of porous system and pore pressures. The physical conditions of ice formation process are described by thermodynamic balance equation between ice, liquid water and solid matrix. Finally, the mechanical response of porous media subjected to the frost action was studied by several authors [4,27,28]. On the one hand, the poroelasticity formulation based on Biot's continuum model was adopted. It is an efficient method for elastic modeling of porous system, which is subjected to the pressure of the fluid. On the other hand, a novel micromechanics approach was introduced to analyze the creation of micro-cracks in the microstructure during freezing process [9,13]. These results predict effective mechanical and transport properties at microscopic level and can be utilized as an input for multi-scale analysis of porous media.
As a preamble, our goal is to quantify the internal damage caused by the ice crystallization pressure in historical mortars. In particular, a critical point in a restoration works is frequent applications of lime mortars for preserving compatibility with the historical materials. However, lime mortars are very porous, their mechanical strength and durability are mostly very low [17,21], thus the development of a lime mortar with improved internal hydrophobicity and associated improved resistance against damage due to the effects of ice crystal-lization is inevitable. To address this issue with respect to its complexity, an analysis combining both experimental work and numerical simulations has to be done. Nevertheless, the numerical methodology developed within this work can be utilized to simulate the response of any porous material subjected to the frost action.
A theoretical formulation of the problem is presented in Section 2, followed by numerical calculations in Section 3. Section 3.1 then investigates the influence of pore size distribution on the evolution of damage parameter. The essential findings are summarized in Section 4.

Material model
The problem of porous system subjected to ice crystallization can be divided into three physical phenomena -heat and moisture transport, ice formation process and evolution of damage caused by pore pressure. Using the thermodynamics, poromechanics and damage mechanics, we propose here the concept of multiphase constitutive model based on the assumption of the uncoupled system in the sense of numerical analysis, see Fig. 1. These models are characterized by combining different physical or mechanical models (in space and time) in order to accurately describe structural response of deteriorating infrastructure over time.
In the presented work, the porous material is treated as multi-phase medium Dirichlet BC:û i+1

Transport model
We use the diffusion model by Künzel, see [10], which is based on Krischer's concept [26]. Künzel neglected the liquid water and water vapor convection driven by gravity and total pressure as well as enthalpy changes due to liquid flow and choose relative humidity ϕ as the only moisture driving force. The water vapor diffusion is then described by Fick's law written as where where is the liquid conductivity and D l [m 2 s −1 ] is the capillary transport coefficient, dw/dϕ is the derivative of water retention function. The Fourier law is then used to express the heat flux q h [Wm −2 ] as where λ [Wm −1 K −1 ] is the thermal conductivity and θ [ • C] is the local temperature. Introducing the above constitutive equations into energy and mass conservation equations we finally get resulting set of differential equations for the description of heat and moisture transfer expressed in terms of temperature and relative humidity as • the energy balance equation • the conservation of mass equation The transport coefficients defining the material behavior are nonlinear functions of the temperature, moisture and material properties. We briefly recall their particular expressions [10]: where w f [kgm −3 ] is the free water saturation and b ϕ [−] is the approximation factor, which must always be greater than one. It can be determined from the equilibrium water content (w 80 ) at 0.8 [-] relative humidity by substituting the corresponding numerical values in Eq. (6). Fig. 2(a) shows an example of variation of water content as a function of relative humidity.
where µ [−] is the water vapor diffusion resistance factor and δ [kgm is the vapor diffusion coefficient in air given by  ). An example of variation of water vapor permeability as a function of temperature is seen in Fig. 2(b).
where D l [m 2 s −1 ] is the capillary transport coefficient given by An example of variation of liquid conductivity D ϕ [kgm −1 s −1 ] as a function of water content is plotted in Fig. 2(c).
where λ 0 [Wm −1 K −1 ] is the thermal conductivity of dry building material, is the bulk density and b tcs [−] is the thermal conductivity supplement. An example of variation of thermal conductivity as a function of water content is shown in Fig. 2(d).
• p sat -water vapor saturation pressure [Pa], where • h v -evaporation enthalpy of water [Jkg −1 ] h v = 2.5008 · 10 6 273.15 θ • H -total enthalpy of porous material [Jm −3 ] For the spatial discretization of the partial differential equations, a finite element method is preferred here to the finite volume technique. The discretized form of energy and moisture balance equations then reads where K is the conductivity matrix, C is the capacity matrix, r is the vector of nodal values, and F is the vector of prescribed fluxes transformed into nodes.
For a detailed formulation of the matrices K and C and the vector F , we refer the interested reader to [23,24].
The numerical solution of the system Eq. (16) is based on a simple temporal finite difference discretization. If we use time steps ∆ t and denote the quantities at time step i with a corresponding superscript, the time-stepping equation is where γ is a generalized midpoint integration rule parameter. In the results presented in this paper the Crank-Nicolson (trapezoidal rule) integration scheme with γ = 0.5 was used. Expressingṙ i+1 from Eq. (17) and substituting into the Eq. (16), one obtains a system of non-linear equations: which can be solved by some iterative method such as Newton-Raphson.

Ice formation process
The ice crystallization process in the porous system is described by the penetration of liquid/ice interface from external surfaces or large pores towards the unfrozen zones [19].
The ice formation process is limited by a critical pore radius r cr (θ), see Fig. 3(a). It describes the smallest geometrical radius of pore in which ice crystal can form [14,28], where r ir [m] is the curvature radius of ice crystal (liquid-ice interface) formed at a given temperature and r ar [m] is the layer of adsorbed water which cannot freeze during crystallization process. The empirical formula for r ar [m] was proposed by Fagerlund [5] as and r ir [m] is introduced through the simplified form of the Gibbs-Duhem equation [27] as where γ li [Nm The concept of the pore pressure caused by the ice crystals is well known, see [3,4,5,14,19,20,22,27,28]. To be more specific, let us consider spherical liquid-ice interface at the entrance of the pore and cylindrical shape of pores.
Two interface equilibrium conditions are assumed to describe the interaction between ice crystals and pore walls. The Laplace relation is applicable to control the interface between the ice crystal and the liquid water: where p l [Pa] is the liquid pressure and p i [Pa] is the pressure in the ice crystal.
The second interface relation is expressed in the form of mechanical equilibrium between the ice crystal and the pore pressure exerted by the ice crystal, p p [Pa], as p i − p p = γ li r − r ar (θ) .
Finally, combining Eq. (22) and Eq. (23), we can write where χ(r, θ) [Pa] is the local pressure on the frozen pore walls due to the ice formation and it is characterized by following relation, see [18,28], It has been advocated in [19] and [28] that the average pore pressure exerted by the ice crystal on the pore walls can be introduced as

Mechanical (damage) model
The description of mechanical behavior of a porous media saturated with a liquid water was firstly proposed by Biot [2] and extended in the more general context of continuum thermodynamics for the ice crystals by Coussy [3,4]. According to this approach, the formula between the effective stress σ ′ [Pa] and total stress σ [Pa] has following form purpose, we utilize more convenient relation for porous system derived in [22] .
where n [−] is the total porosity. As a further extension, we introduce one parameter isotropic nonlocal damage model given by constitutive law, see [1,7], damage law and loading-unloading conditions where where ε I [−] is component of the principal strains and the brackets denote the positive part. According to [1], the local value ε eq [−] is replaced by its nonlocal average defined as about nonlocal formulation we refer to [1]. Finally, the damage law d w = g(κ) is provided by following relation where ε f [−] is the equivalent strain at critical crack opening and ε 0 [−] is the strain at the elastic limit.
Considering the stress relation Eq. (28) and Eq. (30), the linear momentum balance equation for the porous system, can be expressed in the following form:

Boundary and initial conditions
To complete the proposed material model, the initial and boundary conditions are set as follows: • The Dirichlet boundary conditions u =û(t) on Γ I u , • The Neumann boundary conditions • The Robin boundary conditions • Initial conditions  The measured material parameters, corresponding to a lime mortar, are listed in Tab. 1. Several parameters were obtained from a set of experimental measurements providing mostly the hygric and thermal properties of mortar, see [17]. Unfortunately, no additional experiments were conducted for the mechanical properties, hence we utilize some material parameters of mortars mentioned   The results are presented in Fig. 6 showing variation of the temperature and moisture at selected nodes labeled in Fig. 4.   A fast moisture increase in the zone close to the exterior surface ( Fig. 6(b)) leads also to the similar trend of the damage parameter, see Fig. 7

Influence of porosity
The formation of ice is mainly controlled by pore size distribution, see [28]. To address this issue we consider the same input data as in the previous numeri- the value of total porosity changes slightly, while the influence on the evolution of damage parameter is significant. This is clearly observed from the comparison of pore pressures displayed in Fig. 7(a) and Fig. 9(c). Combining all the previous results suggests that the structure of porous system plays crucial role in the resulting pore pressure and subsequently calculated internal damage.

Conclusions
This paper presents the numerical modeling of damage caused by ice crystallization process in mortars. Attention is focused on the thermo-hygro-mechanical model developed here in the framework of uncoupled algorithmic scheme. Two particular issues were addressed: (i) the formulation of material model based on laws of the thermodynamics, poromechanics and damage mechanics, (ii) influ- In particular, we employed Künzel's model, which is sufficiently robust to describe real-world materials, but which is also highly nonlinear, time-dependent material model. Supported by several successful applications in civil engineering we adopted Biot's model and the nonlocal isotropic damage model in the framework to simulate the frost action on porous media.
A crucial point in modeling of damage in mortars is the pore size distribution.
The obtained results suggest a high importance of porosity on evolution of the damage parameter, at least for the present material parameters and applied range of initial and boundary conditions.
Finally a comparison of the numerical calculations with experimental measurements is under current investigation and will be presented elsewhere.