Résumé
L'action des variations d'humidité et de température sur le béton, compte tenu des facteurs de dimension et de forme, ainsi que de la répartition des contraintes est traduite par l'équation fondamentale du fluage, du retrait et de la dilatation thermique. La pâte de ciment et le béton sont étudiés en tant que matériaux composites multiphases dans lesquels les conditions d'équilibre, tant statique que thermodynamique, doivent être considérées.
Summary
The constitutive equation for creep, shrinkage and thermal expansion, which reflects correctly the effect of variable humidity and temperature, including the effect of size, shape and stress distribution, is derived. Cement paste and concrete are treated as a multi-phase composite material, in which both the static and thermodynamic conditions of equilibrium must be considered.
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Abbreviations
- a T,a :
-
rate constants for the local microscopic diffusion in eq. (16), (40) and (45)
- b T,b :
-
rate constants for the macroscopic diffusion in eq. (7)–(8), (45)
- c, C:
-
diffusion constants for adsorbed layers, defined by (7a), (7b)
- \(\bar d\) :
-
average effective distance for microscopic diffusion (volumetric), (eq. 16),
- \(\bar d'\) :
-
similar value for deviatoric diffusion
- e ij :
-
total strain deviator
- e d ij :
-
deviator of the change of thickness\(\bar \delta _d \) of hindered layers
- fa,fd :
-
area factor forpa orpd, respectively (eq. 12, 22)
- h :
-
humidity=relative vapor pressure (in the pores inside)
- h eq :
-
equivalent humidity defined after eq. (49)
- h ex :
-
humidity of the external atmosphere (ambient)
- h s :
-
humidity at self-desiccation of a sealed sample (eq. 5)
- h 0 :
-
time decrease of humidity at T0 for which the volume change without stress is zero (eq. 20)
- k,k′:
-
slopes of the desorption and sorption isotherms (eqs. 3, 5, 41)
- n :
-
exponent in eq. (10)
- p :
-
pressure, less the atmospheric pressure 1 atm
- pa,pv,ps:
-
p in free adsorbed layers (eq. 1), in vapor, or in capillary water (eq. 2a)
- p d :
-
p in the hindered adsorbed layer
- p d :
-
its average value (eqs. 2, 12)
- q :
-
activation energy for hydration (apparent) eq. (52)
- r1,r2 :
-
principal curvature radii of capillary menisci
- s ij :
-
total stress deviator
- s d ij :
-
deviator of the stress in hindered layers
- t :
-
time, or age of concrete
- t e :
-
equivalent curing period, defined after eq. (4)
- u :
-
displacement in the sense ofx
- v :
-
specific volume=(mass density)−1
- va,vd,vc,vv:
-
v for free, or hindered adsorbed layer, capillary water and vapor, respectively
- v c :
-
1 cm3/g
- u :
-
total mass of water (usually per unit volume of porous material or per unit surface)
- we,wn:
-
evaporable and non-evaporable water at a given T
- wa,wd,wc:
-
w corresponding tova,vd,vc
- x, y, z orx1,x2,x3:
-
cartesian coordinates
- B:
-
coefficient of water transmission at the surface, eq. (9)
- G:
-
Gibbs' free energy (eq. A1) or shear modulus (eq. 40)
- G1 :
-
partial value of G per unit of mass
- Ga, Gb, Gc, Gd, Gf, Ka, Kb, Kc, Kd, Kf, Kh:
-
shear moduli and volume moduli for models in figures 6a, 5a (defined by eqs. 23, 24, 27, 40, 18)
- K:
-
elastic volume modulus, eq. (27)
- L:
-
volume stress memory function, eq. (32), (32a) (32e)
- La :
-
humidity memory function, eq. (32b), (32c), (32f)
- LT :
-
temperature memory function, eq. (51), (51a)
- M:
-
molecular weight of water
- Pd :
-
disjoining pressure=Pd— pressure in the free layer of equal thickness (eq. (A5)
- Q, Q′:
-
activation energies (enthalpies) in eq. (45)
- R:
-
universal gas constant
- S:
-
entropy
- T, To :
-
absolute temperature and chosen reference temperature
- V:
-
volume
- αb,αc,αd,αo,α1:
-
thermal expansion coefficients, eq. (47)–(49)
- βT :
-
relative hydration rate defined by eq. (4)
- β:
-
βT at reference temperature, eq. (52)
- βa :
-
volume compressibility of adsorbed water, eq. (14)
- γ:
-
surface tension, eq. (2a) (or shear strain)
- δa, δd:
-
average thickness or free, or hindered adsorbed layer (δa=wa/va, δd=wd/vd)
- \(\bar \delta _d \) :
-
average total thickness of all layers intersecting a unit length (eq. 15)
- ε:
-
total volume strain=(ε11+ε22+ε33)/3
- εij:
-
total strain tensor
- εd :
-
volume component of the change of thickness\(\bar \delta _d \) of hindered layers\(d\varepsilon _d = d\bar \delta _d \)
- ε 50 , εd:
-
free shrinkage (eq. 32b) and creeps train defined before eq. (22a)
- ϰ:
-
hygrothermic coefficient, defined by eq. (41)
- μ:
-
chemical potential (eq. A7)
- σ, γa:
-
total volume stress, and volume stress in the fluid (defined in § 5.2)
- σd, γad:
-
actual volume stress in hindered layers and its theoretical value needed for thermodynamic equilibrium at a givenPa (eqs. 12, 17, 20)
- ϕ, ψ (or\(\tilde \varphi \)):
-
rate of creep constants for volume and deviatoric deformation, eq. (28), (25), (40), (38) (or eq. 29)
- ϕd, ψd :
-
rate constants for microscopic volume and deviatoric diffusion, eq. (17), (19), (37)
- τ,t′:
-
time as integration variable, or time at load application (also τ=shear stress)
- τi:
-
retardation time for thei-th unit in figure 5b=\(\tilde \varphi _i^{ - 1} \); Δ denotes increments during time step Δt; Subscripti infdi, ϕi, Kci, Kdi, Kfi, αadi denotes values offd etc. for thei-th unit in the chain in figure 5b
- kp :
-
kilopond=force kilogram, Å=angström=10−7 mm,\(\dot u\)=(u).=ϱu/ϱt, ∼proportionality sign, ≈ “approximately equal”, → “tending to”, ↓“assign”
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Bažant, Z.P. Constitutive equation for concrete creep and shrinkage based on thermodynamics of multiphase systems. Mat. Constr. 3, 3–36 (1970). https://doi.org/10.1007/BF02475106
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DOI: https://doi.org/10.1007/BF02475106