NISTIR 7265

Effect of continued hydration on the transport properties of cracks through portland cement pastes in a saturated environment: A microstructural model study

K.A. Snyder

Jeffrey W. Bullard
NISTIR 7265

Effect of continued hydration on the transport properties of cracks through portland cement pastes in a saturated environment: A microstructural model study

K.A. Snyder

Jeffrey W. Bullard

Materials and Construction Research Division

Building and Fire Research Laboratory

Sponsor:

U.S. Nuclear Regulatory Commission

August 2005

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. DEPARTMENT OF COMMERCE

Carlos M. Gutierrez, Secretary

TECHNOLOGY ADMINISTRATION

Michelle O’Neill, Acting Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

William Jeffrey, Director
Abstract

The contribution of continued hydration to the healing of cracks in cement paste is

studied using the CEMHYD3D microstructural model. After an initial period of hydration, the computational domain was divided equally into two halves, separated by

water-filled elements, representing a crack, and then hydration continued. The transport

factor of the composite was calculated at various intervals, from which the crack properties were calculated. The crack transport factor was a unique function of crack porosity,

regardless of crack width or cement paste water:cement mass ratio. Minimum crack

porosity at long times depended strongly on the fraction of unhydrated clinker when the

crack was formed. For late-age cracking, the initial rate of crack filling depended upon

the amount of unhydrated clinker exposed by the creation of the crack. In all cases, the

majority of solids in the crack were formed by the redistribution of hydration products

from the paste adjacent to the crack.

Keywords: autogeneous healing; building materials; cement paste; concrete; cracking;

hydration; service life.

iii
Contents

Abstract iii

1 Introduction 1

2 Crack Transport Properties 3

2.1 Formation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Transport Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Influence on Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.2 Influence on Permeability . . . . . . . . . . . . . . . . . . . . . . 4

3 Bulk Properties 5

4 Crack Transport Factor 6

5 CEMHYD3D 7

5.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2 Cement Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.3 Model Paste Formulation and Hydration . . . . . . . . . . . . . . . . . . 8

6 Uncracked Paste Properties 8

6.1 Degree of Hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6.2 Capillary Pore Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6.3 Transport Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Cracked Paste Properties 11

7.1 Transport Factor vs. Time . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7.2 Transport Factor vs. Porosity . . . . . . . . . . . . . . . . . . . . . . . . 12

7.3 Crack Solids Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7.4 Leaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.5 Crack Solids vs. Time: Early Age . . . . . . . . . . . . . . . . . . . . . . 20

7.6 Crack Solids vs. Time: Late Cracking . . . . . . . . . . . . . . . . . . . . 21

7.7 Crack Surface Hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Discussion 24

8.1 Healing Capacity: Early Cracking . . . . . . . . . . . . . . . . . . . . . . 25

8.2 Healing Capacity: Late Age . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.3 Crack Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8.4 Wide Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8.5 Clinker Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8.6 NDE Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 Summary 28

iv
List of Tables

1 Approximate age (days), and corresponding degree of hydration αp, when

paste capillary porosity is no longer percolated. . . . . . . . . . . . . . . 9

2 Degree of hydration at the time of cracking for the three w/c values. . . . 11

3 Crack wall area fraction of unhydrated clinker Aw. The second number

is the expected value calculated from product of the initial solids fraction

and the degree of hydration at the time of cracking. . . . . . . . . . . . . 22

v
List of Figures

1 SEM micrograph of crack through w/c=0.45 cement paste after 28 d hydration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Schematic for the three transport coefficients calculated for each microstructure. There are three calculated coefficients: two parallel with the crack

(Fk), and one orthogonal to the crack (F⊥). . . . . . . . . . . . . . . . . 6

3 Paste degree of hydration αp as a function of time t for CCRL Cement 116.

Experimental results are shown as open symbols and the CEMHYD3D

calculation as a solid line. Large filled circles denote the point when the

capillary pores are no longer percolated. . . . . . . . . . . . . . . . . . . 9

4 Paste transport factor Lp as a function of time t for the three w/c values

hydrated under saturated conditions. . . . . . . . . . . . . . . . . . . . . 10

5 Paste transport factor Lp, divided by the initial value Lo as a function of

paste capillary porosity θp for the three w/c values. . . . . . . . . . . . . 11

6 Crack transport factor Lc as a function of time t for different crack widths

in 0.30 w/c paste. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Crack transport factor Lc as a function of time t for different crack widths

in 0.40 w/c paste. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8 Crack transport factor Lc as a function of time t for different crack widths

in 0.50 w/c paste. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9 Crack transport factor Lc as a function of the crack porosity θc. . . . . . 14

10 Crack transport factor Lc as a function of the reduced crack porosity Θc

(see Eqn. 10). Solid line is Θ3/2

c

. . . . . . . . . . . . . . . . . . . . . . . . 15

11 Crack porosity θc as a function of time t for 2 μm, 4 μm, and 8 μm cracks

initiated at 3.5 d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

12 Ratio of portlandite (CH) to C-S-H in the 0.5 w/c systems at 10,000 cycles,

after developing an 8 μm crack at 500 cycles. Distance x represents slices

parallel to the crack that is located at x = 100; see text for additional

explanation. Inset shows the same ratio for the cracked system, including

the crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13 Porosity voxels of cracked (8 μm at 500 cycles) and uncracked 0.40 w/c

systems at 10 000 cycles hydration. Distance x represents slices parallel to

the crack that is located at x = 100; see text for additional explanation.

Inset shows the difference between the two curves. . . . . . . . . . . . . . 18

14 Leached bulk paste porosity in 0.4 w/c paste having a 8 μm crack initiated

at 3.6 d. Numbers indicate elapsed days after crack formed. . . . . . . . 19

15 Leaching index Λ as a function of the time t after cracking occurs at tc in

systems having w/c = 0.4 and 0.5. The solid lines denote data for cracks

initiated at 3.6 d, and dashed lines denote data for cracks initiated at 33 d. 20

vi
16 Crack solids content for systems crack at 3.5 d, expressed as a function

of crack width w and crack solids fraction φc, as a function of the degree

of hydration αp occurring after the the degree of hydration at the time of

cracking α

i

. The solid lines correspond to 16 μm and 32 μm cracks. The

left and right vertical lines correspond to when the 0.30 w/c paste and the

0.40 w/c paste de-percolate, respectively. The straight line is 5(αp − α

i

). 21

17 Crack solids volume wφc for systems cracked at 33 d, expressed as a function of the degree of hydration αp occuring beyond the degree of hydration

at cracking α

i

. Solid line intersecting the ordinate axis at 0.2 has same

slope as the line in Fig. 16. . . . . . . . . . . . . . . . . . . . . . . . . . . 22

18 Crack solids content wφc, divided by the crack wall area fraction of unhydrated clinker Aw as a function of continued hydration (αp − α

i

). . . . . 23

19 The fraction of the difference between the degree of hydration in the

cracked system αb and that of the uncracked system αp, relative to the

contribution α

w of the clinker exposed at the crack walls. . . . . . . . . . 24

20 Difference between maximum degree of hydration α

max and the timedependent degree of hydration αp as a function of time t. . . . . . . . . . 25

21 Difference between maximum degree of hydration α

max and the paste degree of hydration αp, times the volume fraction of unhydrated clinker in

the paste ψp (see Eqn. 12), as a function of time t. . . . . . . . . . . . . . 26

vii
1 Introduction

The service performance of concrete can be influenced greatly by the presence of cracks.

Therefore, long-term performance prediction for existing concrete structures containing

cracks must consider the time-dependent crack transport properties. The mechanisms by

which crack transport properties can change have been enumerated and classified [1–3]. In

general, the mechanisms act to reduce the crack transport coefficient, effectively healing

the crack.

The majority of mechanisms attributed to crack healing involve the transport of

environmental material through the crack and the accumulation of reaction products

within the crack. These materials may include fine particulates or calcium carbonate

[4]. When these mechanisms operate, the time-dependent crack transport coefficient is

a strong function of the boundary conditions. Therefore, under these circumstances,

assessment of the time-dependent behavior for a particular structure can only be made

if detailed knowledge of the corresponding initial and boundary conditions is available.

Another mechanism for crack healing is the formation of material from continued

hydration [4, 5]. Fortunately, this particular healing mechanism can be isolated, in large

part, from a particular environment. Moreover, the behavior of the system can be determined from the state of the cement paste at the time the crack occurred.

A study of crack healing by continued hydration alone has application to situations

where there is brief water transport at the time of cracking, after which there is no

additional convective flow. Some of these scenarios include water pipes with cementitious

liners, tanks, and young concrete that has cracked because of vibration or shock. [6]

The most common scenario would be healing between freezing cycles in a saturated

environment [7–10]. It has been observed that continued hydration after freeze/thaw

testing has led to partial or complete recovery of the ultrasonic pulse velocity [9, 11]. In

one experiment, continued hydration was attributed to the increase in shear modulus in

samples exposed to freezing and thawing after only 3 d hydration [12].

The present study focuses on continued hydration as a mechanism for crack healing,

thereby reducing the number of experimental parameters. For the scenario considered,

two important parameters are the initial crack volume and the volume of hydration

products that fill the crack. The initial crack volume can be parameterized by the crack

width. The volume of hydration product will depend on the cement paste water:cement

mass ratio (w/c) and the degree of hydration at the time of cracking. If more material

can be produced after cracking, the ultimate crack porosity will be lower.

Although crack width and degree of hydration are well defined laboratory experimental parameters, a computational model is used here in favor of laboratory experiments

for several reasons. First, one objective of this study is to determine the crack transport

properties as a function of time. Although one could make bulk conductivity and permeability measurements, a means would still be needed to unambiguously extract the

crack geometry, which has been shown to have a varying width [13]. Second, the crack

porosity is an important parameter, but is a time-consuming quantity to determine experimentally. A model allows one to eliminate as much of these laboratory complications

as is possible. Moreover, a model study can facilitate a parametric study of principal

components. This is all that is needed to learn about general behavior that is applicable

1
to other systems.

The NIST microstructural development model CEMHYD3D [14] is used here to perform a parametric study of the factors influencing the rate of crack healing due to continued hydration. The hydration model has been developed to accurately represent developing cement paste microstructure at the micrometer length scale, and the digital image

approach facilitates the addition of planar cracks. Moreover, the CEMHYD3D model is

able to mimic cement paste hydration rate [15] and cement paste formation factor [16].

These two factors alone are sufficient for capturing the salient features of crack healing.

Figure 1: SEM micrograph of crack through w/c=0.45 cement paste after 28 d hydration.

The creation of cracks with smooth walls is not an unreasonable approximation for

cracks in cement paste. Figure 1 shows a scanning electron microscope (SEM) micrograph

of a crack propagating through a w/c=0.45 cement paste that is approximately 28 d old.

The crack was generated by a microhardness tester that impacted the sample out of the

field of view. The figure shows that cracks in paste can propagate in reasonably straight

lines, easily passing through hydrated and unhydrated cement with minimal deviation.

This evidence is meant only to justify the use of straight cracks having smooth walls in

this initial study, not as a proof that all cracks are smooth and straight.

For this study, the CEMHYD3D model was modified to accommodate the addition

of cracks during hydration. The initial systems were cement paste cubes having 200 μm

long edges, composed of cubic voxel elements having 1 μm long edges. A crack of a given

width was created by dividing the microstructure into two halves and then displacing

the halves perpendicular to the dividing plane. Intervening elements were filled with

water and hydration continued as before. Periodically, the microstructure was stored as

an image file and the crack porosity and transport properties were calculated from the

image file.

2
2 Crack Transport Properties

There are two crack transport properties of practical concern: diffusivity and permeability. The majority of reported studies on the effect of cracks on bulk transport

have addressed permeability, [1, 4, 5, 17–22] and comparatively few have addressed diffusion/migration [23–25]. The difference in interest is somewhat proportional to the effect

of cracks on each transport property.

Generally, the contribution of cracks to overall transport is the product of the crack

transport property and the area fraction of cracks. In the absence of severe cracking,

the area fraction of cracks is typically expressed in fractions of a percent. Given that

the effective diffusivity of concrete is typically a factor of 10−2smaller than that for bulk

fluid in a crack, the contribution of cracks to overall diffusive transport is usually small.

By contrast, the permeability of a crack has a quadratic dependence on the crack

width. As a result, the permeability of a crack could easily be one million times greater

than that of the bulk concrete. Even after factoring in the crack area fraction, the crack

can still contribute significantly to overall permeability.

2.1 Formation Factor

The formation factor F is a dimensionless number that characterizes the solid microstructure of a random porous media and can be related to the bulk diffusivity and permeability. For a porous solid saturated with a conducting brine, the formation factor F can be

approximated by the ratio of the pore solution conductivity σs to the bulk (solid plus

solution) conductivity σb:

F =

σs

σb

(1)

For a nonconducting solid, this number is strictly greater than or equal to one.

For concrete, the formation factor can be used to characterize either the uncracked

microstructure, the bulk cracked composite, or the crack itself. Generally, laboratory

techniques probe the bulk material (cracked or uncracked) but not the crack itself. In

this paper, however, the formation factor of the crack is expressed independently of the

paste. This will be shown to be an important distinction that leads to greater insight

into the results.

2.2 Transport Factor

A random material can be characterized either by a resistive quantity (analogous to formation factor) or by a conductive quantity, its inverse. Although either can characterize

a material, in the case of transport, it seems more natural to consider the relative contribution to overall transport, which corresponds to the conductivity analogy. Such a

quantity has been used in hydrogeological research [26], and has also been referred to

as the formation factor. Here, for clarity, the conductivity analog is referred to as the

transport factor L:

L =

1

F

(2)

3
2.2.1 Influence on Diffusion

The transport factor is an important parameter for modeling diffusion in porous materials filled with electrolyte [27,28]. Specifically, it is the parameter that relates electrolyte

transport coefficients to bulk transport coefficients. Changes in the transport factor correspond to a proportional change in the effective diffusion coefficient of a particular ionic

species. Therefore, changes in crack transport factor would correspond to proportional

changes in overall bulk diffusivity, with the crack surface area being the coefficient of

proportionality.

2.2.2 Influence on Permeability

The relationship between transport factor and permeability is not as direct as that for

diffusivity. The difficulty arises because permeability depends upon the characteristic

microstructural pore size. The Katz-Thompson [29, 30] (KT) equation, developed for

geophysical research, relates permeability k to the transport factor L:

k =

d

2

c L



(3)

The quantity dc is the microstructural critical pore diameter, and can be estimated from

mercury intrusion porosimetry [29]; it is analogous to the largest diameter sphere that can

pass through the microstructure. The variable  is an empirical constant for a particular

material: e.g., limestone, sandstone, cement paste, etc.

The KT equation can be adapted to the study of cracks by considering the relevant

geometrical parameters. Using purely dimensional arguments, the critical pore diameter

dc within a crack should be a function of the crack width w and the crack porosity θ:

dc = θ

nw (4)

Based on the work of Martys and Garboczi [31] on two-dimensional systems, the exponent

n should be approximately equal to 2 for narrow cracks. If the cracks are considerably

wider than the size of the particles filling the crack, the crack microstructure will be threedimensional, and the critical pore diameter will be larger than that for a two-dimensional

system at the same porosity.

For flat walls separated by a distance w, the permeability kc of a crack with no interior

solids can be calculated assuming Poiseuille flow [32]:

kc =

w

2

12

(5)

Substituting Eqn. 4 into Eqn. 3, and exploiting the exact result in Eqn. 5 in the limit

θ, L → 1, yields a reasonable value of  in Eqn. 3:

kc =

θ

4 w2 L

12

(6)

This relation is used as a starting point for estimating the permeability of healing cracks

as a function of the initial crack width and the crack porosity.

4
Laboratory permeability measurements of cement pastes yield a w

2 dependence in the

permeability, but the constant  is typically greater than 12. This difference is usually

attributed to crack wall surface roughness, and Eqn. 6 is multiplied by a dimensionless

roughness parameter having a value between 0 and 1 [4].

3 Bulk Properties

When studying the effect of cracks on the transport properties of a concrete, it is important to distinguish between the crack transport properties and the bulk transport

properties. Published experiments have determined the transport properties of a slice

taken from a cylinder [19, 20, 22], in which a crack is generated using the split cylinder

test. The bulk transport properties of the sample are determined as a function of the

experimental parameters.

When total flow is measured in the direction along the crack, the total bulk permeability kT is a weighted sum of the crack permeability kc and the concrete permeability

k◦:

kT =

Ackc + A◦k◦

AT

(7)

The weighting factors are the crack cross-sectional area Ac, the concrete cross-sectional

area A◦, and the total cross-sectional area AT = Ac + A◦. For example, a sample taken

from a 100 mm diameter cylinder would have a total cross sectional area of 7,854 mm2.

A 100 μm wide crack that spans the sample diameter will have an area of 10 mm2 and

(from Eqn. 5) a permeability kc of 833 μm2. Assuming a typical concrete permeability

k◦ of 10 nm2[33], the total bulk permeability kT is approximately 1 μm2, a factor of 105

greater than the intact permeability k◦.

Although bulk permeability depends strongly on the presence of cracks, bulk permeability alone is not a sensitive measure of crack permeability. To see why, consider the

previous 100 μm crack. The effect of healing on the total bulk permeability kT can be

expressed relative to the intact permeability k◦:

kT

k◦

≈ 1 +

Ackc

AT k◦

(8)

If, as the crack heals, the bulk permeability kT decreases to within a factor of 2 of k◦,

the crack permeability kc would still be nearly a factor of 800 times greater than the

concrete permeability k◦.

The use of bulk measurements to probe crack diffusivity is even more problematic than

for crack permeability. A fresh crack has, by definition, a transport factor of one. Given

a cement paste transport factor of 0.002, the bulk diffusivity would never be greater than

twice the value for intact paste for the scenario given above. Generally, the transport

factor of concrete would be greater, and the contribution of cracks would be more difficult

to detect.

Because bulk transport properties are not sensitive indicators of the transport properties of cracks, a means is needed to determine crack properties directly. To achieve this

experimentally, one must know precisely the area fraction of the crack and the transport

5
properties of the intact material. In addition, dissolution of mineral phases within the

paste and subsequent diffusion and precipitation within the crack will alter the original

paste transport properties. An account of this effect must be made because, as noted

already, the corresponding changes in bulk transport coefficients are very small.

4 Crack Transport Factor

In this computational experiment, the bulk transport factor is calculated from the threedimensional digitized microstructure image file created by CEMHYD3D. A unit electrical

field is applied in each cartesian direction, and the total current in each direction is used

to calculate the bulk transport factor in each of the three directions. Only water-filled

porosity and calcium silicate hydrate (C-S-H) [34] are assumed to contribute to the overall

conduction. The transport factor used for these two phases are as follows: water: 1.000;

C-S-H: 0.005.

The crack porosity and transport factor must be calculated for each system considered.

Because the crack location within the microstructure is known, the porosity of the crack

is calculated from the number of saturated porosity pixels remaining in the original

crack volume. The crack transport factor, however, can only be calculated indirectly.

One cannot simply extract the crack microstructure from the overall microstructure and

calculate the transport coefficient because the crack walls are porous and these pore play

a role in transport through the crack. Instead, one would have to also extract a portion

of the paste adjacent to the crack. Fortunately, there is an easier approach.

F

l

w

L

L

Figure 2: Schematic for the three transport coefficients calculated for each microstructure. There are three calculated coefficients: two parallel with the crack (Fk), and one

orthogonal to the crack (F⊥).

The transport factor of the paste and crack are calculated from the bulk transport

6
factor by exploiting the symmetry of the problem. A schematic of the calculation is shown

in Fig. 2, where a microstructure has been divided in two and separated by a width w that

corresponds to the crack. The new width of the system is l. The conductivity calculation

yields three bulk transport coefficients: two that are parallel to the crack (Lk), and one

that is perpendicular to the crack (F⊥).

Due to the symmetry of the problem, the CEMHYD3D conductivity calculations in

each direction can be approximated by a corresponding one-dimensional problem composed of two materials (paste and crack), each having homogeneous properties. Parallel

to the crack, the contributions to the bulk transport factor Lk are in proportion to the respective lengths, and perpendicular to the crack, the contributions to the bulk formation

factor F⊥ are in proportion to the respective lengths:

Lk =

l − w

l

Lp +

w

l

Lc

F⊥ =

l − w

l

Fp +

w

l

Fc (9)

The three measured quantities are the two values for Lk and one value for F⊥. The

subscripts p and c represent paste and crack properties, respectively. These equations

were solved twice, once for each value of Lk; the quantity F⊥ was the same in both

calculations. The two solutions yield four values: two values each for Lc and Lp. Because

the two values for Lp were typically within 0.001 % of one another, the three reported

values were the two values for Lc and one of the values for Lp.

5 CEMHYD3D

5.1 Model Description

The principles and underlying assumptions of CEMHYD3D have been described in detail elsewhere [14, 15], so only a brief summary is provided here. CEMHYD3D is a

three-dimensional cellular automaton model, in which the cement paste microstructure

is discretized onto a uniform cubic mesh. Each element of the mesh is filled with some

material (water-filled porosity, alite, etc.). The chemical reactions comprising hydration

are simulated by allowing the elements to interact according to rules that depend on the

materials involved in the interaction. These rules are used to mimic dissolution of solids,

diffusion of dissolved species according to a random walk algorithm, and nucleation and

growth of hydration products such as portlandite and C-S-H gel. Hydration proceeds

in repeated discrete cycles of dissolution, diffusion, and reaction. The elapsed time t

is assumed to depend on the number n of computational cycles executed according to

t = βn2, with a scaling factor β that depends on the cement being simulated. A wide

range of properties can be monitored as hydration proceeds, including heat of hydration,

phase volume fractions, chemical shrinkage, percolation of capillary porosity, and setting

time. Moreover, the microstructure at any time can be input to an auxiliary program to

calculate its transport factor. The model has been applied with considerable success to

elucidate the influence of microstructural and environmental factors on hydration kinetics

and microstructure development [35–39].

7
5.2 Cement Characterization

Because it operates directly on cement paste microstructures, a detailed characterization

of the cement powder is essential for providing accurate input to CEMHYD3D. The data

required by CEMHYD3D are the cement particle size distribution (PSD), the volume

fraction and surface area fraction of each cement phase and, because clinker particles are

composed of multiple phases, the statistical spatial distribution of the clinker phases.

In this study a representative Type I cement was selected. The cement was issued

as Cement 116 by the ASTM-sponsored Cement and Concrete Reference Laboratory

(CCRL), located at the National Institute of Standards and Technology (NIST), as part

of its proficiency sample program [40]. The characterization data for the cement, and a

description of the methods used to obtain the data, have been published elsewhere [15].

5.3 Model Paste Formulation and Hydration

Three-dimensional, digitized microstructures of cement paste, suitable for input to the

CEMHYD3D hydration model, were created using the characterization data for Cement

116. The mesh was 200 elements in each cartesian direction, for a total of 8 × 106

elements. Non-overlapping digitized spherical particles, with sizes following the measured

PSD, were mapped randomly onto the mesh and the interparticle porosity was filled with

water. The total volume of particles was constrained to achieve water:cement mass ratios

of 0.3, 0.4, or 0.5, assuming a density of 3200 kg/m3.

Each microstructure was input to the hydration model, and hydration was simulated

at a constant temperature of 25 ◦C. The paste was assumed to be in contact with a

reservoir of water to replenish water in the capillary pore space consumed by hydration

(until the capillary pore space network became disconnected). Any crack formed during

hydration also was filled with water from the reservoir.

At selected times during hydration, the hydrated microstructure was saved and the

transport factor of the crack and of the bulk paste were calculated using a separate finite

difference program that was designed specifically to operate on digitized microstructures [41].

6 Uncracked Paste Properties

The properties of the uncracked paste, such as degree of hydration, capillary porosity,

and transport factor are presented before the results from the cracking experiment. This

is done to form a point of reference for the data from the cracked pastes. In addition,

the data from the uncracked systems are used to motivate the search for universal relationships between bulk and microstructural properties.

6.1 Degree of Hydration

During numerical hydration, the degree of hydration is calculated from the number of

cement elements consumed by reaction. The uncracked paste degree of hydration αp was

compared to laboratory measurements based on loss-on-ignition (LOI) [15, 42], and the

8
0.1 1 10 100

t (d)

0.0

0.2

0.4

0.6

0.8

1.0

αp

0.50

0.40

0.30

β = 0.00035

Figure 3: Paste degree of hydration αp as a function of time t for CCRL Cement 116.

Experimental results are shown as open symbols and the CEMHYD3D calculation as a

solid line. Large filled circles denote the point when the capillary pores are no longer

percolated.

results are shown in Fig. 3 for all three w/c values. In the figure, laboratory data are

denoted by open symbols, and the values of αp calculated by CEMHYD3D are denoted

by solid curves. The calculated values were adjusted using a constant time parameter

(β = 0.00035) that gave the best agreement between calculated and laboratory values.

6.2 Capillary Pore Percolation

Table 1: Approximate age (days), and corresponding degree of hydration αp, when paste

capillary porosity is no longer percolated.

w/c

0.30 0.40

Age(d) 6.2 26.6

αp 0.56 0.77

As the systems hydrate, the capillary pore space becomes filled with hydration products. For a discrete model such as CEMHYD3D, the time at which the capillary pores no

9
0.1 1 10 100 1000

t (d)

0.001

0.01

0.1

1

Lp

0.50

0.40

0.30

w/c

Figure 4: Paste transport factor Lp as a function of time t for the three w/c values

hydrated under saturated conditions.

longer percolate can be determined uniquely. The point at which this occurs is denoted

by a large filled circle in Fig. 3 for the 0.30 w/c and the 0.40 w/c pastes; the capillary

pores of the 0.50 w/c paste remain percolated indefinitely. The approximate age and

corresponding degree of hydration at which the capillary pores are no longer percolated

are given in Table 1.

6.3 Transport Factor

The time dependence of the cement paste transport factor Lp is shown in Fig. 4 for all

three w/c values. Although the time dependence of the transport factor was qualitatively

similar for all three w/c values, Fig. 5 shows that more universal behavior is revealed

if the data are plotted as a function of the saturated paste capillary porosity θp. This

also has been done previously with a measure of success [16]. Here, in addition, the

transport factor is scaled by its initial value L◦. It is interesting to note that when the

data are presented in this manner, there is no sharp transition in the transport factor as

the porosity falls below 0.20, the point at which it is assumed the capillary pores are no

longer percolated [43, 44].

10
0.1 1

θ

p

0.001

0.01

0.1

1

Lp / L

o

0.30

0.40

0.50

Figure 5: Paste transport factor Lp, divided by the initial value Lo as a function of paste

capillary porosity θp for the three w/c values.

Table 2: Degree of hydration at the time of cracking for the three w/c values.

w/c 3.5 d 33 d

0.30 0.49 0.64

0.40 0.53 0.79

0.50 0.58 0.82

7 Cracked Paste Properties

The results that follow are from systems having a crack width of either 2 μm, 4 μm, 8 μm,

16 μm, or 32 μm. The cracks were initiated either at 3.5 d or 33 d of model hydration.

The degree of hydration of each paste at the time of cracking is shown in Table 2. To

facilitate the analysis, the results for the systems cracked at 3.5 d are presented first. An

analysis of the systems cracked at 33 d is made in context of the methods used to study

the systems cracked at 3.5 d.

As a convenient shorthand, variables having subscripts denote time-dependent quantities, and variables with superscripts denote constants.

11
101 102 103 104

t (d)

0.0

0.2

0.4

0.6

0.8

1.0

L

c

32 μm

16 μm

 8 μm

 4 μm

 2 μm

Figure 6: Crack transport factor Lc as a function of time t for different crack widths in

0.30 w/c paste.

7.1 Transport Factor vs. Time

For each w/c value and crack width, the crack transport factor Lc is plotted as a function

of time in Figs. 6–8. Both values for Lc at each point are plotted in the figures, but often

the two values nearly coincide. For a given w/c value, the transport factors of narrower

cracks decrease more rapidly and reach a lower ultimate value than those for wider cracks.

This trend is expected intuitively because less solid material is required to fill a narrower

crack.

Taken all at once, establishing a consistent picture of crack healing would be a daunting task. Instead, the problem is broken into two separate tasks: determine how the crack

transport factor depends upon crack porosity; and determine the factors that control the

rate that the porosity within the crack is filled with material.

7.2 Transport Factor vs. Porosity

Based on the transport factor calculations for the uncracked paste discussed previously,

there may be a universal relationship between crack transport factor and crack porosity.

For cracks, however, there is no need to divide the crack transport factor by its initial

value because the initial value is one. The crack transport factor Lc for the systems

cracked at 3.5 d are shown in Fig. 9 as a function of the crack porosity θc. As can be

seen in the figure, there appears to be a universal relationship between crack porosity

and crack transport factor, independent of crack width and cement paste w/c value.

12
101 102 103 104

t (d)

0.0

0.2

0.4

0.6

0.8

1.0

L

c

32 μm

16 μm

 8 μm

 4 μm

 2 μm

Figure 7: Crack transport factor Lc as a function of time t for different crack widths in

0.40 w/c paste.

The data plotted in Fig. 9 suggests that there might exist a critical crack porosity,

below which the transport factor Lc is zero. To corroborate this hypothesis, the data in

Fig. 9 are plotted again in Fig. 10 as a function of a reduced crack porosity Θc:

Θc =

(θc − θ

crit)

(1 − θ

crit)

(10)

A log-log plot of the data, using a critical porosity θ

crit equal to 0.35, yields a reasonably

straight line with slope near 1.5. This is used as an approximation for the crack transport

factor Lc:

Lc = Θ3/2

c

(11)

The function Θ3/2

c

is shown as a solid line in Fig. 10, and appears to characterize all the

data.

A value of 0.35 for the critical porosity θ

crit is consistent with results from percolation

studies on two-dimensional systems. Studies of the percolation threshold for randomly

located overlapping circles have yielded critical porosities ranging from 0.31 to 0.38 [45,

46], while pixel-based representations of circles on a lattice have yielded critical porosities

of 0.35 and 0.33 [47], the latter value for higher resolution circles. Similarly, critical

porosities for randomly located overlapping squares of a fixed orientation have ranged

from 0.33 to 0.39 [48–50], and the critical value for the same objects on a lattice was

0.35 [47].

13
101 102 103 104

t (d)

0.0

0.2

0.4

0.6

0.8

1.0

L

c

32 μm

16 μm

 8 μm

 4 μm

 2 μm

Figure 8: Crack transport factor Lc as a function of time t for different crack widths in

0.50 w/c paste.

0 0.2 0.4 0.6 0.8 1

θ

c

0

0.2

0.4

0.6

0.8

1

L

c

0.30

0.40

0.50

Figure 9: Crack transport factor Lc as a function of the crack porosity θc.

14
0.1 1

Θ

c

0.1

1

L

c

0.30

0.40

0.50 Figure 10: Crack transport factor Lc as a function of the reduced crack porosity Θc (see

Eqn. 10). Solid line is Θ3/2

c

.

Another significance of these results is that these model systems having cracks up to

16 μm wide behave like two-dimensional systems. For wider cracks, one would expect

that the microstructure within the crack will have a three-dimensional structure. As such,

these cracks would have a lower critical porosity at percolation. Therefore, the transport

properties of wider cracks would be greater, at the same porosity, that narrower cracks.

The relationship between crack transport factor and crack porosity in the systems

cracked at 33 d were sufficiently similar to the systems cracked at 3.5 d that presenting

the data would be redundant.

7.3 Crack Solids Production

Given the apparent universal relationship between crack porosity and crack transport

factor, the remaining challenge is to determine the time required for a crack to achieve a

given porosity. Crack porosity θc for a 2-μm, a 4-μm, and an 8-μm crack initiated at 3.5 d

is shown in Fig. 11 as a function of time for all three w/c values. Cracks in lower w/c

pastes fill faster, as has been observed experimentally [5]. In seeming contradiction, the

ultimate crack porosity appears to be lower for higher w/c pastes. Note that the porosity

of the 8 μm cracks in the 0.40 w/c paste falls below the porosity of the corresponding

0.30 w/c paste.

Given these two observations, one can develop a conceptual model for crack filling.

The fact that lower w/c pastes fill cracks faster initially means that transport of hydration

15
1 10 100 1000

t (d)

0.4

0.6

0.8

1.0

θ

c

0.30

0.40

0.50

2 μm

4 μm

8 μm

Figure 11: Crack porosity θc as a function of time t for 2 μm, 4 μm, and 8 μm cracks

initiated at 3.5 d.

product from the paste to the crack is not a primary mechanism because the paste

transport factor is considerably smaller for lower w/c pastes. Therefore, crack filling at

early times is probably due to hydration product formed in close proximity of the crack.

At later times, there is a slower crack filling mechanism that seems to favor higher w/c

pastes. This behavior is consistent with leaching of soluble materials in proximity to the

crack. Higher w/c pastes have a greater percentage of portlandite in the pores, and the

higher transport coefficients contribute to additional transport.

7.4 Leaching

The solids within the crack must originate from either hydration of cement elements at

or near the crack wall, or from solids in the paste that dissolve and then precipitate in

the crack (i.e., leaching). Although the microstructural model does not individually label

solids of similar composition, it is still possible to determine what fraction of the solid

elements within the crack originated from the bulk paste. If the crack had no influence

on microstructural development within the bulk paste, the microstructure on either side

of the crack would remain identical to the microstructure of the corresponding uncracked

system. By contrast, any differences between the cracked and uncracked systems in the

bulk paste, particularly the porosity, must be an effect of the crack.

One effect of the crack will be to redistribute portlandite from the paste to the crack,

in a manner similar to leaching. Figure 12 shows the volumetric ratio of portlandite

16
0 50 100 150 200

x

0.1

0.2

0.3

0.4

CH/C-S-H

0 50 100 150 200 0.0

0.5

1.0

1.5

Cracked

Uncracked

Figure 12: Ratio of portlandite (CH) to C-S-H in the 0.5 w/c systems at 10,000 cycles,

after developing an 8 μm crack at 500 cycles. Distance x represents slices parallel to the

crack that is located at x = 100; see text for additional explanation. Inset shows the

same ratio for the cracked system, including the crack.

17
(CH) to C-S-H, calculated from the voxels in the 200×200 planes parallel to the crack.

The system is the 0.5 w/c paste after 10,000 cycles, with a 8 μm crack appearing after

500 cycles. The main plot shows both the cracked and uncracked systems, with the

crack being omitted. The inset graph shows the ratio for the cracked system, including

the crack. Far away from the crack, the CH/C-S-H ratios are nearly equal for both

systems. In the paste near the crack the ratio decreases, and inside the crack the ratio

is considerably higher than that for the paste. These two effects are consistent with CH

leaching from the paste and precipitating inside the crack.

0 50 100 150 200

x

5000

10000

15000

Porosity Voxels

0 50 100 150 200

0

2000

4000

6000

8000

Cracked

Uncracked

Figure 13: Porosity voxels of cracked (8 μm at 500 cycles) and uncracked 0.40 w/c

systems at 10 000 cycles hydration. Distance x represents slices parallel to the crack

that is located at x = 100; see text for additional explanation. Inset shows the difference

between the two curves.

Other soluble species in the model would also be undergoing the same phenomenon,

leading to an overall increase in the paste porosity in proximity to the crack. Figure 13

shows the number of porosity pixels in each 200×200 plane parallel to the crack; the crack

planes have been omitted from the comparison. The data in the figure demonstrate that

the number of porosity pixels in the paste is nearly the same for both the cracked and

uncracked microstructures, except within approximately 15 μm of the crack, where the

porosity in the cracked system increases. The difference in the two curves in Fig. 13 is

shown in the inset. Therefore, not only are voxels of soluble material leaching from the

paste, they are leaving behind regions of higher porosity just outside the crack.

The total number of porosity pixels in a plane P is calculated for both the cracked

(Pc) and uncracked systems (Po). The difference (Pc − Po) is plotted in Fig. 14 for the

18
0 50 100 150 200

n

0

2000

4000

6000

8000

P

c

 - P

o

 11

 55

 230

 521

1455

day

Figure 14: Leached bulk paste porosity in 0.4 w/c paste having a 8 μm crack initiated

at 3.6 d. Numbers indicate elapsed days after crack formed.

0.4 w/c paste having a crack of width 8 μm. The numbers in the figure denote the

elapsed time in days since cracking first occurred. Because the degree of hydration of

the cracked systems was always within 2 % of that for the uncracked system, the paste

portion of these systems after the same number of cycles should be nearly identical. The

data show random fluctuations about zero for the paste farthest from the crack, and a

dramatic increase in the number of porosity pixels in proximity to the crack.

The sum of (Pc − Po) represents the total number of solid elements missing from the

paste portion of the cracked system. Because these calculations are made on a closed

system, these solid elements could only have diffused into the crack. If Σp represents the

sum of (Pc − Po), and Sc represents the number of solid elements in the crack, the ratio

Σp/Sc represents the fraction of solids in the crack that can be accounted for by leaching

from the paste. Here, this ratio will be referred to as the leaching index Λ.

The leaching index Λ for the 0.40 w/c and the 0.50 w/c cement pastes are shown in

Fig. 15. The data for the 0.40 w/c system includes only the systems cracked at 3.5 d,

while the data for the 0.50 w/c system includes data for both the 3.5 d cracking and the

33 d cracking. Difficulties in how CEMHYD3D handles systems with very low porosity

prohibited a quantitative analysis of the 0.30 w/c systems. The results for the 0.40 w/c

and 0.50 w/c pastes indicate that the majority of solids in the crack can be attributed

to leaching from the bulk cement paste, regardless of when cracking occurs.

Although the effect of the crack is to leach portlandite from the surrounding paste, the

mechanism of leaching in the microstructural model is not physically accurate. There

19
0 200 400 600 800 1000

t - t

c

 (day)

0.2

0.4

0.6

0.8

1.0

Λ

w/c = 0.40

0 200 400 600 800 1000

t - t

c

 (day)

0.2

0.4

0.6

0.8

1.0

Λ

w/c = 0.50

Figure 15: Leaching index Λ as a function of the time t after cracking occurs at tc in

systems having w/c = 0.4 and 0.5. The solid lines denote data for cracks initiated at

3.6 d, and dashed lines denote data for cracks initiated at 33 d.

is no rigorous model for transport within the model. Rather, there is dissolution of

portlandite, and the portlandite "diffuses" as a solid species that has some probability

of precipitating. This is true for all the soluble minerals in the model. Therefore, the

amount of leaching is related to other microstructural development parameters like the

degree of hydration.

7.5 Crack Solids vs. Time: Early Age

Both the volume of hydration product formed and the volume of soluble materials leached

into the crack should be (nearly) independent of the crack width. Therefore, the total

volume of solids within a crack, at a given time, should be independent of crack width.

The volume of hydration product that can be formed after cracking occurs depends upon

the difference between the degree of hydration at the time of cracking and the maximum

degree of hydration. Moreover, this same difference characterizes the amount of materials

that can be leached from the cement paste.

To test this assertion, the crack solids content, expressed as the product of crack

width w and crack solids fraction φc = (1 − θc), is plotted in Fig. 16 as a function of

the difference between the paste degree of hydration αp and the degree of hydration at

the time of crack initiation α

i

(see Table 2). The dotted and dashed curves are labeled,

and the solid lines correspond to 16 μm and 32 μm cracks. The solid straight line passes

through the origin and has a slope of 5. Although, for a given w/c paste, the curves for

all the crack widths do not lie upon one another, the curves for the wider crack widths

are in proximity to one another. Given the discrete nature of the CEMHYD3D model,

the deviations for narrower cracks are attributed to finite-size effects.

It is interesting to note that both the 0.30 w/c and the 0.40 w/c paste curves in Fig. 16

diverge above the straight line, while the 0.50 w/c curve diverges below the straight line.

A common distinction between these two groups is that the 0.30 w/c and the 0.40 w/c

pastes experience a de-percolation of the capillary porosity during the calculation. The

amount of hydration after cracking (αp − α

i

) at which this occurs is shown in Fig. 16 as

20
0 0.1 0.2 0.3 0.4

α

p

 - α

i

0

0.5

1

1.5

2

2.5

w 

φ

c

(

μm)

2 μm

4 μm

8 μm 0.30 w/c

0.50 w/c

0.40 w/c

Figure 16: Crack solids content for systems crack at 3.5 d, expressed as a function of

crack width w and crack solids fraction φc, as a function of the degree of hydration αp

occurring after the the degree of hydration at the time of cracking α

i

. The solid lines

correspond to 16 μm and 32 μm cracks. The left and right vertical lines correspond to

when the 0.30 w/c paste and the 0.40 w/c paste de-percolate, respectively. The straight

line is 5(αp − α

i

).

the two vertical lines. The divergence of the 0.30 w/c and the 0.40 w/c pastes from the

straight line occurs near the time at which the capillary pores are no longer percolated,

which suggests that the de-percolation of capillary porosity causes the diverging behavior.

7.6 Crack Solids vs. Time: Late Cracking

A similar analysis was performed for the systems cracked at 33 d, and the data are

shown in Fig. 17. The straight line in the figure has a slope equal to that shown in

Fig. 16. Although the relationship between crack solids content and degree of hydration

after cracking eventually become somewhat linear with similar slopes as for the systems

cracked at 3.5 d, there is a period immediately after cracking during which the filling

mechanism is noticeably different, as is apparent by the different initial slopes for the

different w/c values. This different behavior is likely due to a crack filling mechanism

that has not yet been discussed.

At 33 d, each system already has hydrated to more than 88 % of its ultimate degree

of hydration. Therefore, when a crack forms at this age, the contribution of continued

bulk hydration must be significantly less than that for the systems cracked at 3.5 d.

21
0 0.02 0.04 0.06 0.08 0.1

α

p

 - a

i

0

0.2

0.4

0.6

0.8

w 

φ

c

(

μm)

2 μm

4 μm

0.30 w/c

0.50 w/c

0.40 w/c

Figure 17: Crack solids volume wφc for systems cracked at 33 d, expressed as a function

of the degree of hydration αp occuring beyond the degree of hydration at cracking α

i

.

Solid line intersecting the ordinate axis at 0.2 has same slope as the line in Fig. 16.

By comparison to Fig. 16, when cracks form in older pastes, less hydration occurs and

considerably less hydration product is formed in the crack. This has been observed

experimentally [9].

Table 3: Crack wall area fraction of unhydrated clinker Aw. The second number is the

expected value calculated from product of the initial solids fraction and the degree of

hydration at the time of cracking.

w/c 3.5 d 33 d

0.30 0.229/0.238 0.151/0.168

0.40 0.172/0.190 0.066/0.085

0.50 0.140/0.147 0.054/0.064

Based on the model used here, the process of creating cracks exposed unhydrated

clinker at the crack wall. This sudden exposure to unhydrated clinker could, at these

late ages, contribute significantly to the volume of crack solids. Table 3 shows the crack

wall area fraction of unhydrated clinker Aw, along with an estimate of the same quantity

based on the initial cement volume fraction and the degree of hydration at the time of

cracking. In general, the two values agree with one another.

22
0 0.02 0.04 0.06 0.08 0.1

α

p

 - a

i

0

2

4

6

8

10

w 

φ

c

 / A

w

(

μm)

2 μm

4 μm

0.30 w/c

0.50 w/c

0.40 w/c

Figure 18: Crack solids content wφc, divided by the crack wall area fraction of unhydrated

clinker Aw as a function of continued hydration (αp − α

i

).

Given the advanced degree of hydration for the systems cracked at 33 d, it may be

possible for the exposed unhydrated clinker on the crack wall to contribute significantly

to the initial rate of crack filling. Based on this hypothesis, the crack solids volume

wφc, divided by the crack wall area fraction of clinker Aw, is plotted as a function of the

degree of hydration after cracking, and is shown in Fig. 18 for all three w/c values. The

co-linearity near the origin among the different w/c values suggests that crack surface

hydration may indeed play an important role in the initial rate of crack filling when

cracks appear at late ages.

7.7 Crack Surface Hydration

Because CEMHYD3D is a discrete model, the contribution of crack wall surface hydration

can be, to a certain extent, extracted from the degree of hydration of the cracked system

αb. As the cracked system hydrates, the rate of hydration will be slightly greater than

that for the uncracked system. The difference could be both from crack surface cement

hydration and from additional hydration in the bulk due to the presence of the water

in the crack. Although the individual hydration sites were not monitored during the

calculation, it may be possible to make some general inferences based on knowing the

number of crack surface cement elements.

In CEMHYD3D, the degree of hydration is calculated from the number of volume

elements of cement remaining at any given moment. Therefore, at the time of crack

23
formation, there exists a number of cement elements on the crack surface. The ratio of the

number of cement elements on the crack surface to the total number of cement elements

initially corresponds to the contribution to the overall degree of hydration attributed to

hydration of the first layer of cement element on the crack surface α

w.

To quantify the contribution of crack wall hydration, it is compared to the difference

between the degree of hydration in the cracked system αb and the degree of hydration in

the uncracked paste αp. This difference, divided by the potential contribution to crack

surface hydration α

w, would correspond to the equivalent number of crack surface layers

that would have to hydrate to account for the difference.

0 0.02 0.04 0.06 0.08 0.1

α

p

 - a

i

0

1

2

3

4

5

6

7

(

α

b - 

αp) / 

α

w

0.30 w/c

0.40 w/c

0.50 w/c

Figure 19: The fraction of the difference between the degree of hydration in the cracked

system αb and that of the uncracked system αp, relative to the contribution α

w of the

clinker exposed at the crack walls.

These data are shown in Fig. 19 for the systems cracked at 33 d. As can be seen

in the figure, one equivalent layer of crack surface cement elements has hydrated after

(αp − α

i

) = 0.02 for all three w/c values. Interestingly, the same value for (αp − α

i

) in

Fig. 18 is near the transition between the initial production mechanism (initial slopes)

and the final production mechanism (final slopes).

8 Discussion

Although one cannot state definitively the exact crack healing behavior of all cement

pastes based on this model study, one should be able to argue for general relationships

24
and for principle components (i.e., those physical parameters having the greatest influence

on a result).

8.1 Healing Capacity: Early Cracking

For systems developing cracks at early ages, the minimum crack porosity achievable

depends upon the degree of hydration at the time of cracking, along with the maximum

degree of hydration. Based on data shown in Fig. 16, the difference between the maximum

degree of hydration α

max and the degree of hydration at the time of cracking αi determines

the maximum solids content of a crack.

0.1 1 10 100 1000

t (d)

0.0

0.2

0.4

0.6

0.8

1.0

α

max

 - 

αp

0.30

0.40

0.50

Figure 20: Difference between maximum degree of hydration α

max and the timedependent degree of hydration αp as a function of time t.

A plot of (α

max − αp) is shown in Fig. 20 for all three w/c values. This plot is a

measure of the healing capacity of a crack, when cracks occur at early ages. Generally,

higher w/c value pastes have a greater capacity to fill in cracks of a given width. Based

on the curves for the 0.40 w/c and 0.50 w/c pastes, however, the capacity appears to

saturate because of the competition between rate of hydration and maximum degree of

hydration α

max

.

8.2 Healing Capacity: Late Age

At later ages, the volume fraction of unhydrated clinker may play a role in crack filling,

which suggests an alternative interpretation of late age healing capacity. The volume

25
10 100 1000

t (d)

0.00

0.01

0.02

0.03

ψp (

α

max

 - 

αp

)

0.30

0.40

0.50 Figure 21: Difference between maximum degree of hydration α

max and the paste degree

of hydration αp, times the volume fraction of unhydrated clinker in the paste ψp (see

Eqn. 12), as a function of time t.

fraction of unhydrated clinker in the paste ψp depends on the paste degree of hydration

αp and the initial volume fraction of cement in the paste ψ

◦

p

:

ψp = ψ

◦

p

(1 − αp) (12)

The product ψp(α

max − αp) is plotted in Fig. 21 as a function of time for the three w/c

values. Based on this definition of healing capacity, low w/c pastes should heal better

when cracking occurs at late ages.

The validity of this definition of healing capacity, suggesting that 0.4 w/c pastes

should perform better for early cracking, is partially supported by Fig. 11. Although

initially the porosity decreases fastest for the 0.3 w/c paste, it appears that the porosity

of the corresponding 0.4 w/c and 0.5 w/c pastes eventually fall below that of the 0.3 w/c

paste. For increasing crack width, however, the porosity of the 0.40 w/c paste fell below

that of the 0.30 w/c paste much sooner than the 0.50 w/c paste. Therefore, although

the 0.40 w/c and the 0.50 w/c pastes should have roughly the same healing capacity

for early cracks, the presence of more unhydrated clinker in the 0.40 w/c paste has the

practical effect of speeding the recovery of the 0.40 w/c paste over that of the 0.50 w/c

paste. These results suggest that if cracks are anticipated at early age, a moderate w/c

is best, and if cracks are anticipated at later ages, a lower w/c is best.

26
8.3 Crack Permeability

The relationship between crack transport factor and crack porosity given in Eqn. 11 can

be substituted into the general expression for the permeability of pseudo two-dimensional

structures given in Eqn. 6 to yield an approximation for the permeability of a crack kc

as a function of the crack initial width w and porosity θc:

kc = θ

4

c

 

θc − 0.35

1 − 0.35 !3/2

w

2

12

(13)

Based on the observed data, a crack porosity of 0.5 appears to be lower bound on

hydration-induced crack healing. At this porosity, Eqn. 13 predicts the crack permeability would decrease to approximately 1/144 of its initial value. If the formation of the

cracks increased the overall permeability by a factor of γ, reducing the crack porosity

to 50 % would reduce the overall permeability to approximately γ/144. Therefore, one

would see approximately a two order of magnitude decrease in the overall permeability.

Of course, this minimum porosity could only be achieved in the smallest cracks.

8.4 Wide Cracks

The widths of cracks used in this study were limited by experimental constraints. The

2003system is near the practical limit of existing computational resources. Given this

dimensional constraint, cracks that are wider than 32 μm would consume a significant

fraction of the total volume and would contribute to a majority of the total system

porosity. Regardless, the data suggest that calculations using wider cracks would not

contribute significantly to the understanding of continued hydration on crack healing.

By observation of the results in this study, and by extrapolation, one could conclude

that cracks that are significantly wider than approximately 10 μm will probably not heal

much, if at all, due to continued hydration alone. Under the most ideal conditions of early

cracking in a 0.4 w/c paste, Fig. 16 suggests an upper bound for wφc to be approximately

2.5 μm. For the case of a 100 μm crack, the maximum crack solids fraction would be

2.5 %, or 97.5 % porosity. Using Eqn. 13, the hydration product would reduce the crack

permeability to 85 % of the water-filled crack permeability. It is likely that this change

would be within experimental uncertainty, and the one might conclude that healing is

insignificant, or nonexistent, in cracks wider than 100 μm. This conclusion is consistent

with experimental observations of a diminishing number of crack filling “bridges” as the

crack width approaches 10 μm [23]. In field concrete, however, other mechanisms can

contribute to the filling of wider cracks, but they may be more difficult to quantify.

8.5 Clinker Aggregates

Some of the inferences made from this experiment are predicated on the expectation that

a crack forming in the paste content of a concrete will expose surfaces of unhydrated

cement. This generally should happen if the cohesive strength of the C-S-H is greater

than either the cohesive strength of the unhydrated material or the adhesive strength

of the interface between C-S-H and the unhydrated cement particles on which it forms.

27
However, in the absence of fresh cement surface being exposed at newly-formed crack

walls, the effect of the unhydrated clinker volume on late-age crack healing could be

practically non-existent.

One could, in some applications, improve the chances of exposing unhydrated cement

by using clinker or coarsely ground cement as a component of the aggregates. The use of

clinker as aggregate has been proposed previously [51], but not for application to crack

healing. As shown in Fig. 1, cracks propagate through unhydrated cement. Therefore,

cracks should propagate through more coarsely ground cement because they still have

the defects that grinding eliminates. Although the addition of clinker aggregate would

improve the overall crack healing performance for any w/c paste, based on the results

presented here, the right formulation of clinker aggregate in a 0.40 w/c mixture could

have superior performance at all ages of cracking.

8.6 NDE Significance

A unique relationship between crack transport factor and crack porosity, as suggested

by the data in Fig. 9, would have a very practical significance for crack characterization

in the field. In general, the three crack parameters that have the greatest influence on

overall transport, in decreasing order of significance, are crack depth, crack transport

coefficient, and crack width. The first two can effect bulk transport properties by orders

of magnitude. For fixed crack depth and crack transport coefficient, however, the total

transport through the crack is only linearly dependent on the crack width.

Fig. 9 suggests that determining crack porosity is sufficient for characterizing the

crack transport coefficient, regardless of the crack width. Given corroborating evidence,

it would suggest that NDE research on characterizing cracks could achieve great gains

by developing the ability to estimate crack porosity.

9 Summary

A microstructural model can be a useful investigatory tool for evaluating the principal

components having the greatest effect on the healing of cracks through continued hydration. Results indicate that the transport property of a crack depends largely on the

crack porosity, independent of initial crack width or cement paste w/c value. This result

is consistent with the premise that the mechanism by which the cracks fill is independent

of crack width and cement paste w/c. This result is significant for future investigations

attempting to characterize cracks using non-destructive evaluation (NDE) techniques.

Further analysis of the results suggests that the mechanism by which the cracks heal

has two components. The initial rate of healing depends upon the amount of unhydrated

cement exposed at the time of cracking. These hydration reactions lead to a rapid

production of hydration products within the crack. The initial hydration of exposed

unhydrated cement is quickly exhausted, and the remainder of the crack filling depends

upon redistribution of hydration product from the bulk paste. For cracking occurring at

early ages, the contribution to crack filling from the hydration of exposed unhydrated

clinker is very small. At later ages, however, the contribution of exposed unhydrated

28
clinker can become comparable to that of continued hydration. This result led to the

speculation of the advantage of using clinker fine aggregate to mitigate the effect of

cracking at late ages.

Acknowledgements

The authors would like to thank Paul Stutzman (National Institute of Standards and

Technology) for providing the SEM micrograph in Fig. 1.

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