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Modeling cementitious materials as multiphase porous media: theoretical framework and applications

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Abstract

In this paper a general model for the analysis of concrete as multiphase porous material, based on the so-called Hybrid Mixture Theory, is presented. The development of the model equations, taking into account both bulk phases and interfaces of the multiphase system is described, starting from the microscopic scale. An exploration of the second law of thermodynamics is also presented: it allows defining several quantities used in the model, like capillary pressure, disjoining pressure or effective stress, and to obtain some thermodynamic restrictions imposed on the evolution equations describing the material deterioration. Then, two specific forms of the general model adapted to the case of concrete at early ages and beyond and to the case of concrete structures under fire are shown. Some numerical simulations aimed to prove the validity of the approach adopted also are presented and discussed.

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Abbreviations

a αβ :

area of αβ interface per averaging volume

A α :

specific Helmholtz free energy for the bulk phase α (J kg−1)

A αβ :

specific Helmholtz free energy for the interface αβ (J kg−1)

a αβ :

specific surface of the αβ-interface (m−1)

C p :

effective specific heat of porous medium (J kg−1 K−1)

\({C_p^{\rm g}}\) :

specific heat of gas mixture (J kg−1 K−1)

\({C_p^{\rm w}}\) :

specific heat of liquid phase (J kg−1 K−1)

D :

total damage parameter (–)

d :

mechanical damage parameter (–)

d α :

strain rate tensor of the bulk phase α (s−1)

\({{\bf D}_{\rm d}^{\rm gw}}\) :

effective diffusivity tensor of water vapor in dry air (m2 s−1)

E:

Young’s modulus (Pa)

E0 :

Young’s modulus of mechanically undamaged material (Pa)

E s :

strain tensor for the solid phase (finite strain)(–)

E α :

internal energy of bulk phase α (J kg−1)

E αβ :

internal energy of interface αβ (J kg−1)

\({\hat{e}_{\alpha \beta}^\alpha}\) :

rate of mass transfer to the bulk phase α from interface αβ (kg m−3 s−1)

\({\hat{e}_{\alpha \beta \gamma}^{\alpha \beta}}\) :

rate of mass transfer to the interface αβ from the contact line αβγ (kg m−3 s−1)

e s :

elastic strain for the solid phase (infinitesimal deformations) (–)

g :

gravity acceleration (m s−2)

g αβ :

acceleration of the αβ interface (m s−2)

Hα :

enthalpy of the bulk phase α (J kg−1)

Hαβ :

enthalpy of the interface αβ (J kg−1)

hα :

heat source in the bulk phase α (W kg−1)

hαβ :

heat source on the interface αβ (W kg−1)

I :

unit tensor (–)

\({J_{\alpha \beta}^\alpha}\) :

curvature of the αβ interface with respect to the bulk phase α (m−1)

\({{\bf J}_d^{\rm gw}}\) :

diffusive flux of vapor (kg m−2 s−1)

\({{\bf J}_d^{\rm ga}}\) :

diffusive flux of dry air (kg m−2 s−1)

k :

absolute permeability tensor (m2)

k :

absolute permeability (scalar) (m2)

k rπ :

relative permeability of π-phase (π = g,w) (–)

\({\dot {m}_{\rm dehydr}}\) :

rate of mass due to dehydration (kg m−3 s−1)

\({\dot {m}_{\rm vap}}\) :

rate of mass due to phase change (kg m−3 s−1)

n :

total porosity (pore volume/total volume) (–)

p c :

capillary pressure (Pa)

p g :

pressure of gas phase (Pa)

p w :

pressure of liquid water (Pa)

p s :

solid skeleton pressure (Pa)

p ga :

dry air partial pressure (Pa)

p gw :

water vapor partial pressure (Pa)

q α :

heat flux vector for the bulk phase α (W m−2)

q αβ :

heat flux vector for the interface αβ (W m−2)

\({\hat {Q}_{\alpha \beta}^\alpha}\) :

body supply of the heat to the bulk phase α from the interface αβ (W m−3)

\({\hat {Q}_{\alpha \beta \gamma}^{\alpha \beta}}\) :

body supply of the heat to the interface αβ from the contact line αβγ (W m−3)

Q :

fourth order tensor for the definition of transient thermal strain (–)

S w :

liquid phase volumetric saturation (liquid volume/pore volume) (–)

\({{\hat{\bf S}}_{\alpha \beta \gamma}^{\alpha \beta}}\) :

body supply of momentum to the αβ-interfaces from the αβγ-contact line (kg m−2 s−2)

s αβ :

stress tensor for the αβ-interface (Pa m)

T :

absolute temperature (K)

T α :

absolute temperature of the α-phase (K)

T αβ :

absolute temperature of the αβ-interface (K)

T cr :

critical temperature of water (K)

T max :

maximum temperature attained during dehydration process (K)

t :

time (s)

t s :

Cauchy stress tensor of the solid phase (Pa)

t α :

partial stress tensor of the α-phase (Pa)

t Total :

total stress tensor of the multiphase system (Pa)

\({{\hat {\bf T}}_{\alpha \beta}^\alpha}\) :

Body supply of momentum to the bulk phases from the interfaces (kg m−2 s−2)

u :

displacement vector of solid matrix (m)

V :

thermo-chemical damage parameter (–)

v α :

velocity of the α-phase (m s−1)

v α, s :

relative velocity of the α-phase (with respect to the solid phase “s”) (m s−1)

v α, αβ :

relative velocity of the α-phase (with respect to αβ-interface) (m s−1)

v gs :

relative velocity of gaseous phase (m s−1)

v ws :

relative velocity of liquid phase (m s−1)

w αβ :

velocity of the interface αβ (m s−1)

w αβ, s :

relative velocity of the αβ-interface (with respect to the solid phase “s”) (m s−1)

\({x_{\rm s}^{\rm ws}}\) :

solid surface fraction in contact with the wetting phase (water) (–)

\({x_{\rm s}^{\rm gs}}\) :

solid surface fraction in contact with the non-wetting phase (gas) (–)

α :

generic bulk phase

α :

Biot’s constant (–)

α c :

convective heat transfer coefficient (W m−2 K−1)

αβ :

generic interface of α-and β-phases

β tchem :

thermo-chemical strain coefficient (K−1)

β c :

convective mass transfer coefficient (m s−1)

β s :

cubic thermal expansion coefficient of solid (K−1)

β swg :

combine (solid + liquid + gas) cubic thermal expansion coefficient (K−1)

β sw :

combine (solid + liquid) cubic thermal expansion coefficient (K−1)

\({\bar {\beta}_{\rm tr}}\) :

normalized transient thermal strain coefficient (K−1 s)

β w :

thermal expansion coefficient of liquid water (K−1)

χ eff :

effective thermal conductivity (W m−1 K−1)

ΔHvap :

enthalpy of vaporization per unit mass (J kg−1)

ΔHdehydr :

enthalpy of dehydration per unit mass (J kg−1)

ε sh :

shrinkage strain tensor (–)

ε tchem :

thermo-chemical strain tensor (–)

ε th :

thermal strain tensor (–)

ε tot :

total strain tensor (–)

ε tr :

transient thermal strain tensor (–)

\({{\bar{\boldsymbol \varepsilon}}_{\rm tr}}\) :

normalized transient thermal strain (–)

\({\tilde {\varepsilon}}\) :

equivalent strain in damage theory of Mazars (–)

Γαβ :

surface excess mass density of αβ-interface (kg m−2)

Γhydr :

degree of hydration (or dehydration) (–)

γ αβ :

macroscopic interfacial tension of the αβ-interface (J m−2)

η α :

volumetric fraction of the α phase (–)

μ π :

dynamic viscosity of the constituent π-phase (π = g, w) (Pa s)

λ α :

specific entropy of the α-phase (J kg−1 K−1)

λ αβ :

specific entropy of the αβ-interface (J kg−1 K−1)

Λα :

rate of net production of entropy of the α-phase (W m−3 K−1)

Λαβ :

rate of net production of entropy of the αβ-interface (W m−3 K−1)

Λ :

stiffness matrix of damaged material (Pa)

Λ o :

stiffness matrix of undamaged material (Pa)

Πw :

disjoining pressure (Pa)

ρ :

apparent density of porous medium (kg m−3)

ρ α :

density of the α phase (kg m−3)

ρ g :

gas phase density (kg m−3)

ρ w :

liquid phase density (kg m−3)

ρ s :

solid phase density (kg m−3)

ρ ga :

mass concentration of dry air in gas phase (kg m−3)

ρ gw :

mass concentration of water vapor in gas phase (kg m−3)

\({\hat {\Phi}_{\alpha \beta}^\alpha}\) :

body entropy supply to the bulk phase α from the interface αβ (W m−3 K−1)

\({\hat {\Phi}_{\alpha \beta \gamma}^{\alpha \beta}}\) :

body entropy supply to the interface αβ from the contact line αβγ (W m−3 K−1)

τ s :

Bishop effective stress tensor of the skeleton (Pa)

\({\tilde{{\boldsymbol{\tau}}}^{\rm s}}\) :

“net” effective stress tensor of the skeleton (Pa)

\({\frac{\partial}{\partial t}}\) :

partial time derivative

grad:

gradient operator (spatial description)

div:

divergence operator (spatial description)

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Authors and Affiliations

  1. Department of Structural and Transportation Engineering, University of Padua, via Marzolo 9, 35-131, Padua, Italy

    F. Pesavento & B. A. Schrefler

  2. Department of Building Physics and Building Materials, Technical University of Łódź, Al. Politechniki 6, 93-590, Łódź, Poland

    D. Gawin

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  1. F. Pesavento
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  2. D. Gawin
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  3. B. A. Schrefler
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Correspondence to F. Pesavento.

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Dedicated to Professor Wilhelm Schneider on the occasion of his 70th birthday

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Pesavento, F., Gawin, D. & Schrefler, B.A. Modeling cementitious materials as multiphase porous media: theoretical framework and applications. Acta Mech 201, 313–339 (2008). https://doi.org/10.1007/s00707-008-0065-z

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