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Spherical and acicular representation of hydrates in a micromechanical model for cement paste: prediction of early-age elasticity and strength

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Abstract

Early-age stiffness and strength evolution of cement paste is studied in the framework of continuum micromechanics. Based on the self-consistent scheme, elastic and strength properties are upscaled from the scale of several micrometers up to the scale of several hundreds or thousands of micrometers. Four material phases are considered: clinker, hydration products, water and air. We assign a spherical geometry to clinker grains and pores, while we investigate both spherical and acicular (needle-type) shapes as geometrical representation of the micrometer-sized hydration products. As regards macroscopic poromechanical boundary conditions, two extreme cases are considered: drained conditions and sealed conditions, respectively. These choices allow for studying the influence of (i) the morphological representation of hydrates, and of (ii) the bulk stiffness of water, on the micromechanical prediction of early-age behavior of cement paste, including setting and the hydration-dependent evolutions of both elastic stiffness and uniaxial compressive strength. The newly proposed strength model is based on a von Mises-type elastic limit criterion for individual hydrates. Corresponding deviatoric stress peaks within hydrates are estimated through quadratic stress averages. In this way, the micromechanical strength criterion is formulated in terms of macroscopic loading (stresses or strains, respectively). Model-predicted elasticity and strength evolutions are compared with data from experimental testing of cement pastes with water–cement ratios ranging from 0.35 to 0.60. Satisfactory agreement between model predictions and experiments allows for two conclusions: the morphology of hydrates significantly influences micromechanics-based elastic stiffness estimates of cement paste particularly at very early ages, whereas elastic properties of mature cement paste can be estimated reliably on the basis of both spherical or acicular shaped hydrates. The development of a reliable strength model, however, requires consideration of hydrates as non-spherical particles, no matter what age of cement paste is considered.

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Abbreviations

C 2 S :

abbreviation for dicalcium silicate 2CaO · SiO2 2

C 3 A :

abbreviation for tricalcium aluminate 3CaO · Al2O2 3

C 4 FA :

abbreviation for calcium aluminoferrite 4CaO·Al2O3 · Fe2O3

C 3 S :

abbreviation for tricalcium silicate 3CaO·SiO2

C-S-H:

abbreviation for calcium silicate hydrates

\({\mathbb {C}}\) :

elastic stiffness tensor (fourth-order tensor)

\({\mathbb {C}_p}\) :

\({\mathbb {C}}\) of material phase p

\({\mathbb {C}^{\rm hom}}\) :

macroscopic (homogenized) elastic stiffness tensor (fourth-order tensor), self-consistent estimate

\({{C}^{\rm hom}_{ijrs}}\) :

ijrs-th component of \({\mathbb {C}^{\rm hom}}\)

dV :

infinitesimal volume element

E :

macroscopically prescribed, uniform strains (second-order tensor)

E hom :

self-consistent estimate of macroscopic Young’s modulus

f :

frequency of ultrasonic waves

f air :

volume fraction of air

f c :

uniaxial compressive strength

f clin :

volume fraction of cement clinker

f hyd :

volume fraction of hydration products

fH2O:

volume fraction of water

f p :

volume fraction of material phase p

\({\mathbb {I}}\) :

symmetric fourth-order unity tensor

i :

index

\({\mathbb {J}}\) :

volumetric part of \({\mathbb {I}}\)

j :

index

k hom :

self-consistent estimate of macroscopic bulk modulus

k air :

bulk modulus of air

k clin :

bulk modulus of cement clinker

k hyd :

bulk modulus of hydration products

kH2O:

bulk modulus of water

k p :

bulk modulus of material phase p

\({\mathbb {K}}\) :

deviatoric part of \({\mathbb {I}}\)

n :

number of summation terms

\({\mathbb {P}_{\mathcal{S}_p}^{\rm scs}}\) :

self-consistent expression of Hill tensor for material phase p with phase particles of morphological shape \({\mathcal{S}_p}\) (fourth-order tensor)

\({\mathbb {P}_{\rm sph}^{\rm scs}}\) :

\({\mathbb {P}_{\mathcal{S}_p}^{\rm scs}}\) for a material phase p constituted by spherical particles

\({\mathbb {P}_{\rm cyl}^{\rm scs}}\) :

\({\mathbb {P}_{\mathcal{S}_p}^{\rm scs}}\) for a material phase p constituted by acicular particles

RVE:

abbreviation for representative volume element

p :

index denoting a specific material phase

q :

index denoting a specific material phase

r :

index, scalar parameter

s :

index, scalar parameter

t :

scalar parameter

\({\underline {u}}\) :

displacement vector

V L :

compressive (longitudinal) wave velocity

\({V^{\rm hom}_{L}}\) :

self-consistent estimate of V L

V T :

shear (transverse) wave velocity

\({V^{\rm hom}_{T}}\) :

self-consistent estimate of V T

W :

elastic energy stored in the RVE

w/c :

water-cement ratio

\({\underline {x}}\) :

position vector labeling locations within the RVE and on its boundary

z :

substitution for cos \({\vartheta}\)

α :

parameter depending on both k hom and μ hom, defined in Eq. (25)

β :

parameter depending on both k hom and μ hom, defined in Eq. (25)

δ ij :

Kronecker delta

∂Ω :

surface of the RVE

ε :

linearized strain tensor

ε dev :

deviatoric part of ε

\({\epsilon^{\rm dev}}\) :

deviatoric strain measure, defined in Eq. (9)

\({\underline{\zeta}}\) :

direction vector

ζ i :

component of \({\underline{\zeta}}\)

\({\overline{\overline{\epsilon}}^{\rm dev}_p}\) :

quadratic average of \({\epsilon^{\rm dev}}\) over material phase p

\({\vartheta}\) :

Euler angle (spherical coordinate)

λ :

wave length of ultrasonic waves

μ hom :

self-consistent estimate of macroscopic shear modulus

μ air :

shear modulus of air

μ clin :

shear modulus of cement clinker

μ hyd :

shear modulus of hydration products

μH2O:

shear modulus of water

μ p :

shear modulus of material phase p

ξ :

degree of hydration

ρ air :

mass density of air

ρ clin :

mass density of cement clinker

ρ cp :

macroscopic mass density of cement paste

ρ hyd :

mass density of hydration products

ρH2O:

mass density of water

Σ :

macroscopically prescribed, uniform stresses (second-order tensor)

σ hyd :

second-order stress tensor of hydrates

\({\sigma_{\rm hyd}^{\rm dev}}\) :

deviatoric part of σ hyd

\({\sigma_{\rm hyd}^{\rm dev}}\) :

deviatoric stress measure for hydrates, defined in Eq. (28)

\({\overline{\overline{\sigma}}^{\rm dev}_{\rm hyd}}\) :

quadratic average of \({\sigma_{\rm hyd}^{\rm dev}}\)

\({\sigma_{\rm crit}^{\rm dev}}\) :

critical value of \({\sigma_{\rm hyd}^{\rm dev}}\)

φ :

Euler angle (spherical coordinate)

Ω :

volume of the RVE

Ω hyd :

subvolume of Ω, occupied by hydrates

\({\Omega_{\rm hyd}^{\vartheta,\varphi}}\) :

subvolume of Ωhyd, occupied by acicular hydrates with orientation defined by \({\vartheta}\); and φ

Ω p :

subvolume of Ω, occupied by material phase p

ω :

scalar weight

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  1. Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Karlsplatz 13/202, 1040, Vienna, Austria

    Bernhard Pichler, Christian Hellmich & Josef Eberhardsteiner

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  1. Bernhard Pichler
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  2. Christian Hellmich
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Pichler, B., Hellmich, C. & Eberhardsteiner, J. Spherical and acicular representation of hydrates in a micromechanical model for cement paste: prediction of early-age elasticity and strength. Acta Mech 203, 137–162 (2009). https://doi.org/10.1007/s00707-008-0007-9

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