ECC derives its tensile ductility by deliberately allowing cracks to grow out from pre-existing flaws in a controlled manner. The pre-existing flaws may be a result of naturally entrapped air bubbles or may be deliberately implanted as artificial flaws. Unlike the design of ultra high strength concrete, ECC benefits from the pre-existing flaws by managing their size range rather than by eliminating them. Typically, the damage pattern under uniaxial tension is a large number of closely spaced microcracks of width less than 100 μm, spaced a few millimeters apart. In this manner, localized catastrophic brittle fracture is suppressed. The ductility of ECC is the sum total of distributed deformation resulting from the diffused microcrack damage.
2.1 Strength Criterion
The strength criterion states that the tensile stress σ
cr to initiate a crack from a pre-existing flaw must be below the bridging capacity of the fibers σ
0 crossing that crack. That is,
$$ \sigma_{{cr}} < \sigma_{o} $$
(1)
Satisfaction of this strength criterion assures that the initiated crack will not result in catastrophic loss of load carrying capacity on this crack plane. The matrix cracking strength σ
cr
is dependent on the initial flaw size c and the matrix fracture toughness K
m
. In most cases, the matrix fracture toughness can be assumed to be uniform throughout the material, but flaw sizes do vary in space. As a result, crack initiation starts at the largest and most favorably oriented flaw and progressively works its way to smaller and smaller flaws as the tensile load increases. In other words, the microcrack density typically increases with increasing tensile load. Crack spacing decreases with each additional microcrack formed.
On each crack plane, the load shed by the matrix is taken over by the bridging fibers. The load carried by the bridging fiber is a function of the opening of the crack, characterized by a σ(δ) relationship that increases to a peak and then decreases. The peak value σ
0, termed as the fiber bridging capacity, varies from one crack plane to another due to the inevitable spatial non-uniformity in fiber dispersion. If Eq. (
1) is violated on any given crack plane, that crack localizes into a fracture that terminates the multiple cracking process. If this occurs on the very first crack, only one crack will be present followed by a tension-softening process. This is the typical case for ordinary fiber reinforced concrete. Very small flaws with σ
cr
higher than the lowest σ
0 on already existing crack planes will never be activated.
Clearly, to maintain (
1) in the face of material variability, a high margin or difference between σ
0 and σ
cr
is preferred. In order words, a high (σ
0/σ
cr
) ratio is desired (Kanda and Li
2006). Low σ
cr
can be attained by lowering the matrix fracture toughness
K
m
or by having large flaw size
c. However, excessive lowering of
K
m
or large flaw size
c could lead to low first crack strength and compressive strength. A more preferred scenario is to enhance σ
0. This can be achieved with a high fiber volume fraction
V
f
, a strong fiber, and/or strong fiber/matrix bond. A high fiber volume fraction, however, implies high cost, poor workability, and inhomogeneous fiber dispersion. A strong fiber typically incurs higher material cost. A strong fiber/matrix bond can be engineered; however, excessively high bond leads to fiber rupture that violates the energy criterion, as explained below. Maintaining the strength criterion for multiple cracking while satisfying first cracking and compressive strength, cost, and workability requirements represents a delicate balance when designing ECC (Li et al.
2001,
2002).
2.2 Energy Criterion
After a crack initiates from a pre-existing flaw, the manner in which this crack propagates dictates whether the bridging fibers will be ruptured or pulled out. A lot of this depends on the opening magnitude of the crack. If crack opening is large, rupturing or pulling out of fibers on the crack flanks will be inevitable as the crack propagates. In an ideally brittle material, the crack opening is known to scale parabolically with distance behind the crack tips, which means that the crack opening will increase indefinitely with the propagating crack length. In normal fiber reinforced concrete, the parabolic shaped opening near the crack tip is modified by the closing pressure of bridging fibers, but otherwise follows the same trend in terms of crack opening scaling with the crack length. This implies that the bridged crack must undergo tension softening as fibers are ruptured or pulled out in such composites. This is the normal scenario for ordinary fiber reinforced concrete.
In ECC, fiber rupture or pullout is carefully controlled by restricting the opening of the propagating cracks, typically to below 100 μm. To achieve this, a different mode of crack propagation is required. This is the steady state flat crack propagation mode in which crack opening is uniform at δ
ss
except for a small region behind the crack tip. This flat crack propagation mode requires a balance of energy so that the work done due to applied tensile load σ
ss
on the body must equate to the energy required to break down the crack tip material
J
tip
and the energy required to open the crack against fiber bridging from 0 to δ
ss
. This concept can be summarized as
$$ \sigma_{{ss}} \delta_{{ss}} - \int\limits_{0}^{{\delta_{{ss}} }} {\sigma (\delta )d\delta = J_{{tip}} } $$
(2)
where
$$ J_{{tip}} = \frac{{K_{m}^{2} }}{{E_{c} }} $$
(3)
and
E
c
is the composite Young’s Modulus. The left hand side of (
2) represents the area to the left of the σ(δ) relation at σ
ss(δ
ss), often abbreviated as the complementary energy. Since the σ(δ) relation goes through a peak σ
0(δ
o), the complementary energy has the maximum value of
$$ J_{b}^{'} \equiv \sigma_{o} \delta_{o}- \int\limits_{0}^{\delta_{o} } {\sigma (\delta )d\delta } $$
(4)
To ensure flat crack propagation mode, therefore,
J
b
′
must exceed
J
tip
, or
$$ J_{b}^{'} > J_{tip} $$
(5)
Equation (
5) represents the energy criterion for multiple cracking. A high ratio of
J
b
′
/J
tip
is preferred for robust multiple cracking in the presence of material variability (Kanda and Li
2006).
J
tip
is the energy equivalent of the matrix fracture toughness
K
m
.
J
b
′
is directly related to the fiber and fiber/matrix interface properties. Detailed expressions for σ(δ) relationships have been previously derived for fibers and interfaces with various characteristics (Lin and Li
1997). Most importantly, Eq. (
4) indicates that excessively low or high interfacial bond is not preferred since it suppresses the
J
b
′
value. In other words, the selection of fiber and the control of interfacial parameters are paramount to robust multiple cracking and tensile ductility in ECC.
The dual criteria for multiple cracking provide systematic guidance for designing ECC. Specifically, a low matrix fracture toughness K
m
(or J
tip
), appropriate flaw size c, and a high complimentary energy J
b
′
are conducive to multiple cracking.
K
m
and c are properties of the matrix material. They are influenced by the binder and aggregate types and size, w/c ratio, and other factors. Typically, a more reactive binder (e.g. cement vs mineral admixtures) would lead to a higher K
m
, as would a given binder with finer particle size (more reactive surfaces). Larger aggregate size tends to increase K
m
due to the creation of a more tortuous crack path. For both binder and aggregates, the consideration is to limit the energy barrier to microcrack initiation and propagation. The initial flaw size distribution is also a function of matrix material design. Typically, flaws of a few mm exist due to trapped air during mixing, and may be influenced by workability (especially when fibers are included). Some chemical admixtures also generate air bubbles. However, if the correct range of natural flaw sizes is not present, artificial means to introduce them can be considered. In other words, flaws in ECC are to be managed, not eliminated. For matrix composition in ECC, therefore, the binder, aggregate, and flaw size population are the most important design parameters.
The (maximum) complimentary energy
J
b
′
is a property of the fiber and fiber/matrix interface. High fiber strength is necessary to maintain a high σ
0 without fiber rupture. High fiber stiffness is helpful to maintain tight crack width but may not be conducive to high
J
b
′
. High fiber aspect (length to diameter) ratio is preferred; although this is limited by workability as mixing with high aspect ratio fibers tends to create fiber balling and inhomogeneous dispersion. Excessively high aspect ratio also may lead to excessive amount of fiber rupture. Perhaps the most important factors controlling
J
b
′
are the fiber/matrix interfacial properties—specifically the chemical and frictional bonds. Contrary to intuition, high chemical bond which limits debonding of fiber from matrix is not preferred as it suppresses the
J
b
′
value. Instead, a high frictional bond during fiber slippage is advantageous to maintaining adequate load carrying capacity across the multiple cracks while allowing crack opening to occur in a controlled (non-catastrophic) manner. In ECC, the interface is designed to have a relatively low chemical bond, and the frictional bond actually increases with slippage, i.e. slip-hardening. This can be accomplished by an appropriate amount of wettability of the fiber surface in combination with appropriate densification in the interfacial transition zone (Li et al.
2002). In the case of excessively hydrophilic surfaces, a thin coating could be used to introduce wettability. In the case of fibers with hydrophobic surfaces, it may be necessary to enhance bonding of the fiber to cement via polar group implantation using techniques such as plasma treatment (Wu and Li
1999). Tuning the fiber’s mechanical and geometric properties and the fiber/matrix interface properties for a given matrix is a key element in ECC design.
From the above discussions, it is clear that choosing (or designing) the right fiber is important in attaining multiple cracking and tensile ductility in ECC. What is often overlooked is the equally important matrix and fiber/matrix interface controls. Having a high performance fiber combined with a poorly matching matrix and/or interface will lead to an ordinary fiber reinforced concrete with single crack tension-softening behavior.