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Environmentally motivated modeling of hygro-thermally induced stresses in the layered limestone masonry structures: Physical motivation and numerical modeling

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Abstract

In the present paper, environmentally motivated numerical modeling for the limestone layered masonry structures has been investigated involving the continuous hygric state field variable, i.e. relative humidity Π. By taking advantage of the Lin et al. assumption pertaining to the relative humidity (independency of the relative humidity and temperature) (Lin MW et al. Build. Environ. 41(5):646–656, (2006); Khoshbakht M et al. Finite Elem. Anal. Des. 42(5):414–429, (2006); Khoshbakht M, Lin MW Meas. Sci. Technol 2989–2996, (2006); Khoshbakht M Finite Elem. Anal. Des., 45(8–9):511–518, (2009)), we have provided a mathematical model involving hygro-thermo-mechanical aspects as well as the water vapor transfer across the porous limestone masonry walls. The numerical study substantiates the impact of hygric effects as the major key point in the thermo-hygro-mechanical degradation and effect of geometry in the real brick-line of mortar assembly. Furthermore, we have obtained the moisture entrapment at the intersection of the lines of mortar through the layered masonry wall by means of the multidisciplinary nonlinear finite element method (NFEM) for variably saturated porous media. The new outlooks and fresh departure in durability and aging have been briefly discussed.

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Abbreviations

\({\bar{T}_0=273.15}\) :

Fixed reference temperature in [K]

λ:

First Lamé’s constant in [Pa or N/m2]

Λ g :

Gas degree of saturation in [m3/m3]

\({\mathfrak{K}_s}\) :

Intrinsic permeability or absolute permeability and dynamic viscosity in [m2]

μ :

Second Lamé’s constant in [Pa or N/m2]

ν :

Poisson’s ratio in [−]

Φ g :

Gas porosity in [m3/m3]

\({\rho^{\mathcal {H}_l}}\) :

Liquid water density in [kg/m3]

\({\rho^{\mathcal {H}_{sv}}}\) :

Saturated water vapor density in [kg/m3]

ρ s :

Solid mass density in [kg/m3]

τ :

Tortuosity in [m/m]

Θ0 :

Reference or initial volumetric moisture content in \({\left[\frac{{\rm m}^3}{{\rm m}^3}\right]}\)

A :

Mass uptake coefficient in \({[{\rm kg/m}^2/\sqrt{{\rm Sec.}}]}\)

B :

Visible water level uptake coefficient in \({[{\rm m}/\sqrt{{\rm Sec.}}]}\)

c :

Coefficient of molecular diffusivity of water vapor into air in [m2/Sec.]

\({C^{\mathcal{P}}}\) :

Specific heat capacity at constant pressure in [J/kg/K]

\({C^{\mathcal{V}}}\) :

Specific heat capacity at constant volume in [J/kg/K]

\({C^{{\mathcal{P}}_l}}\) :

Specific heat capacity of liquid water at constant pressure in [J/kg/K]

\({C^{{\mathcal{P}}_v}}\) :

Specific heat capacity of water vapor at constant pressure in [J/kg/K]

D a :

Molecular diffusivity of water vapor into air in [kg/m3]

E :

Modulus of elasticity in [Pa or N/m2]

\({E^\parallel}\) :

Tuffeau modulus of elasticity along parallel direction to the sedimentary bed in [MPa]

\({E^\perp}\) :

Tuffeau modulus of elasticity along normal direction to the sedimentary bed in [MPa]

g :

Gravity acceleration in [m/Sec.2 or N/kg]

h fg (T):

Latent heat (entropy) of vaporization in [J/kg]

K :

Bulk modulus in [Pa or N/m2]

\({k^{\mathcal{T}_{\rm dry}}}\) :

Dry thermal conductivity coefficient in [W/m/K]

\({k^{\mathcal{T}_{\rm psat}}}\) :

Partially saturated thermal conductivity coefficient in [W/m/K]

\({k^{\mathcal{T}_{\rm sat}}}\) :

Fully saturated thermal conductivity coefficient in [W/m/K]

K s :

Permeability coefficient of a saturated porous body in [m/Sec.]

L f :

Latent heats of fusion in [J/kg]

n 0 :

Exponent coefficient of molecular diffusivity of water vapor into air in [−]

P 0 = 101.325:

Reference pressure in [kPa]

\({R^{\mathcal{H}_l}}\) :

Water gas constant in [J/kg/K]

T 0 :

Reference or initial temperature in [K]

\({{\sigma_c}^\parallel}\) :

Compressive strength along normal direction to the sedimentary bed in [MPa]

\({{\sigma_c}^\perp}\) :

Compressive strength along normal direction to the sedimentary bed in [MPa]

\({\mathbb{M}^{3\times3}}\) or \({\mathbb{R}^3 \times \mathbb{R}^3}\) :

Set of real 3 ×  3 second-rank tensors

\({\mathbb{N}}\) :

Natural set

\({\mathbb{R}}\) :

Real set

\({\mathbb{R}^+}\) :

Real positive set

\({\mathbb{R}^-}\) :

Negative real set

\({\mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3}\) :

Set of real fourth-rank tensors

\({\beta^{\mathcal{C}}}\) :

Chemical-induced second-rank stress tensor in [Pa or N/m2]

\({\beta^{\mathcal{H}}}\) :

Moisture-induced second-rank stress tensor in [Pa or N/m2]

\({\beta^{\mathcal{M}_p}}\) :

Plastic-induced second-rank stress tensor in [Pa or N/m2]

\({\beta^{\mathcal{T}}}\) :

Thermal-induced second-rank stress tensor in [Pa or N/m2]

\({\epsilon:={\rm sym} \nabla \otimes u={\rm sym} \nabla u}\) :

Total infinitesimal strain in [−]

\({\hat{\epsilon}}\) :

First-order strain tensor approximation in [−]

\({{1\!\!\!\:1 }:=\delta_{ij}\hat{e}_i \otimes \hat{e}_j}\) :

Identity matrix in [−]

\({\nabla \otimes u:=\nabla u=u_{j,i}\hat{e}_i \otimes \hat{e}_j}\) :

Displacement gradient in [−]

σ :

Symmetric stress tensor in [Pa or N/m2]

\({\tt{F}}\) :

Gradient deformation tensor in [−]

\({\tt{F}^{\mathcal{C}}}\) :

Decomposed gradient deformation tensor indicating chemical-based stretch in [−]

\({\tt{F}^{\mathcal{H}}}\) :

Decomposed gradient deformation tensor indicating hygric stretch in [−]

\({\tt{F}^{\mathcal{M}_e}}\) :

Decomposed gradient deformation tensor indicating elastic stretch in [−]

\({\tt{F}^{\mathcal{M}_p}}\) :

Decomposed gradient deformation tensor indicating plastic stretch in [−]

\({\tt{F}^{\mathcal{T}}}\) :

Decomposed gradient deformation tensor indicating thermal stretch in [−]

ξ :

Tensor of relative humidity swelling coefficient in [−]

A :

Initial symmetric stress tensor in [Pa or N/m2]

F(ΔΠ):

Relative humidity swelling expansion function in [−]

G(ΔΘ):

Moisture swelling expansion function in [−]

\({k^{\mathcal {H}_l}}\) :

Tensor of hydraulic conductivity coefficient in [m2/Pa/Sec.]

\({k^{\mathcal {H}_l}(P)}\) :

Tensor of hydraulic conductivity coefficient as a function of liquid water pressure in [m2/Pa/Sec.]

\({k^{\mathcal{T}^*}}\) :

Thermal conductivity tensor coefficient with no moisture Movement in [W/m/K]

\({k^{\mathcal{T}}}\) :

Thermal conductivity tensor coefficient in [W/m/K]

\({L:=v_{i,j}\, \hat{e}_i \otimes \hat{e}_j}\) :

Asymmetric velocity gradient tensor in [1/Sec.]

X and Y :

Arbitrary second-rank tensors

\({{\alpha}^{\mathcal{T}}}\) :

Tensor of thermal expansion coefficient in [1/K]

\({{\epsilon}^{\mathcal{C}_{\rm cryst}}}\) :

Crystallization strains made by the crystallization pressure in [−]

\({{\epsilon}^{\mathcal{C}_{\rm diss}}}\) :

Dissolution-based strain in [−]

\({{\epsilon}^{\mathcal{C}_{\rm exo}}}\) :

Exothermic strain in [−]

\({{\epsilon}^{\mathcal{C}_{\rm leach}}}\) :

Calcium leaching strain in [−]

\({{\epsilon}^{\mathcal{C}}}\) :

Chemical-based strain in [−]

\({{\epsilon}^{\mathcal{H}}}\) :

Moisture-induced strain or hygric strain in [−]

\({{\epsilon}^{\mathcal{M}_e}}\) :

Elastic strain in [−]

\({{\epsilon}^{\mathcal{M}_p}}\) :

Plastic strain in [−]

\({{\epsilon}^{\mathcal{T}}}\) :

Thermal strain in [−]

\({{\zeta}^{\mathcal{T}}}\) :

Tensor of moisture swelling expansion coefficient in [−]

ΔM :

Uptake mass during water uptake phenomenon or liquid water imbibition in [kg]

\({\dot{Q}}\) :

Heat source/sink rate in [J or N.m]

\({\dot{W}}\) :

Work done rate by control volume in [J or N.m]

η(T):

Dynamic viscosity in terms of temperature in [Pa.Sec.]

\({\hat{\Pi}}\) :

First-order relative humidity approximation in [−]

\({\hat{T}}\) :

First-order temperature approximation in [K]

Λ:

Degree of saturation in [−]

\({\mathfrak{b}}\) :

Fluid body force scalar potential function in [N/m2]

\({\mathfrak{C}(\Delta \Pi)}\) :

Isotropic chemical strain function in [−]

\({\mathfrak{H}(\Delta \Pi)}\) :

Isotropic hygric strain function in [−]

Φ:

Capillary porosity in [−]

Π:

Relative humidity in [−]

ψ :

Matric head or hydraulic head in [m]

ρcT :

Volumetric internal energy in [J/m3]

\({\rho^{\mathcal {H}_v}}\) :

Vapor density in [kg/m3]

ρ s e :

Internal energy per volume involving the moisture migration in [J/m3]

\({\sigma^{\mathcal{H}_l}}\) :

Surface tension of liquid water in [N/m]

\({\tt{h}}\) :

Visible water uptake level in [m]

Θ:

Volumetric moisture content or so-called moisture content in \({[\frac{{\rm m}^3}{{\rm m}^3}]}\)

\({\Theta^{\mathcal {H}_l}}\) :

Volumetric liquid water content or volumetric moisture content in [m3/m3]

\({\Theta^{\mathcal {H}_v}}\) :

Volumetric water vapor content in [m3/m3]

\({\varrho e}\) :

Volumetric internal energy in [J/m3]

\({\varrho Q}\) :

Heat source/sink in [J/m3]

\({\varrho S}\) :

Fluid source/sink in [kg/m3]

\({\varrho}\) :

Fluid density in [kg/m3]

A 0 :

Initial strain energy density in [J/m3 or N.m/m3]

k r (Λ):

Relative permeability in [−]

N(x 1, x 2, x 3):

Quadratic iso-parametric shape function in [−]

P :

Matric suction or liquid pressure in [N/m2]

P v :

Air-water vapor mixture pressure or so-called water vapor pressure in [Pa]

Q Diff :

Diffusion-induced heat source or sink in [J/m3]

Q Exo :

Exothermic heat source or sink in [J/m3]

Q Reac :

Chemical reaction-induced heat source or sink in [J/m3]

Q So/Si :

Heat source or sink in [J/m3]

\({S^{\mathcal {H}_l}}\) :

Liquid source in the considered control volume in [kg/Sec./m3]

\({S^{\mathcal {H}_v}}\) :

Liquid source condensation in the considered control volume in [kg/Sec./m3]

T :

Temperature function in [K]

\({T^{\mathcal{SP}}}\) :

Temperature for a single gas-filled pore in [K]

\({V^{\mathcal {H}_l}}\) :

Liquid water volume in [m3]

\({V^{\mathcal {H}_v}}\) :

Vapor volume in [m3]

\({W_{\tt{mp}} \left(\epsilon^{\mathcal{M}_e}\right)}\) :

Strain energy density in [J/m3 or N.m/m3]

\({{\bar{\nu}}^{\mathcal{H}_l}}\) :

Kinematic viscosity of liquid water in [?]

\({\rho^{\mathcal {H}_l} v^{\mathcal {H}_l}}\) :

Liquid water flux in [kg/m2/Sec.]

\({\rho^{\mathcal {H}_v} v^{\mathcal {H}_v}}\) :

Water vapor flux in [kg/m2/Sec.]

\({\rho^{\mathcal{H}_l} {\hat{v}}^{\mathcal{H}_l}+\rho^{\mathcal{H}_l}{\hat{v}}^{\mathcal{H}_l}}\) :

First-order moisture flux vector approximation in [kg/m2/Sec.]

ρ s b :

Body force in [N/m3]

ρ s f :

Body inertia in [N/m3]

υ :

Fluid flow velocity in [m/Sec.]

υ rel :

Relative velocity in [m/Sec.]

\({\varrho b}\) :

Fluid body force in [N/m3]

a, b and x :

Arbitrary vectors

H :

Potential gradient vector indicating the gravitational effects in [Pa/m or N/m3]

n :

Outward unit normal vector in [−]

\({q^{\mathcal{T}}}\) :

Heat flux rate including the moisture migration hypothesis in [W/m2]

t (n) := σ T · n :

Surface traction or stress vector in [Pa or N/m2]

u :

Displacement vector in [m]

\({V^{\mathcal {H}_l}}\) :

Darcy’s law macro-scale velocity of liquid water in [m/Sec.]

\({V^{\mathcal {H}_v}}\) :

Macro-scale velocity of water vapor in [m/Sec.]

\({{\hat{q}}^{\mathcal{T}}}\) :

First-order heat flux vector approximation in [W/m2]

υ cv :

Control volume velocity in [m/Sec.]

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Authors and Affiliations

  1. CRMD, UMR CNRS 6619- Research Center on Divided Materials, École Polytechnique de l’Université d’Orléans, 8 rue Léonard de Vinci, 45072, Orléans cedex 2, France

    Hamidréza Ramézani

  2. ESTP/IRC/LM-Lean Modeling-École Spéciale des Travaux Publics, du Bâtiment et de l’industrie (ESTP), 28 avenue du Président Wilson, 94234, Cachan cedex, France

    Jena Jeong

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  1. Hamidréza Ramézani
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  2. Jena Jeong
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Correspondence to Hamidréza Ramézani.

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Ramézani, H., Jeong, J. Environmentally motivated modeling of hygro-thermally induced stresses in the layered limestone masonry structures: Physical motivation and numerical modeling. Acta Mech 220, 107–137 (2011). https://doi.org/10.1007/s00707-011-0463-5

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  • DOI: https://doi.org/10.1007/s00707-011-0463-5

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