Page 7 - ICJ Jan 2026

TECHNICAL PAPER

results in the creation of new surfaces, i.e., interfaces between factors affecting strength. For example, at macro level, the DOE
a solid and a surrounding fluid, usually air. Creation of every guidelines in the UK present an infinite number of strength-w/c
such interfacial surface requires energy to overcome cohesive curves for normal strength concrete [12-14] . Each curve represents a
forces that hold the material together. Hence, surface energy specific concrete at a given age. These curves can be expressed
must be supplied during fracture to create new surface. This
energy is supplied by the applied strain energy. For flawless through Equation 6, (Abrams’ Law) in terms of strength f 0.5 for
w/c = 0.5. It implies that aggregate, cement etc. all have a role
homogeneous material, this strain energy required to cause in strength [13,14] .
fracture would be large, but in practical materials such as glass
containing blow holes, or cement paste/concrete containing (6)
pores the energy required would be relatively less as stress
concentration at the tip of a flaw intensifies the local strain where, and
energy supply triggering fracture at a lower average stress. This The equations were obtained through regression in two
concept was applied to explain fracture of plain concrete by stages-first linear equation involving ln(f c ) and w/c were fitted
[7]
Glucklich in the 1960s . Fracture may initiate around multiple to estimate slope ln(K 2 ) and intercept ln(K 1 ) for all the curves
pores, however, several researchers also investigated concrete reported in the reference literature . The values of f 0.5 at
[11]
fracture around a notch in 1960s using linear elastic fracture w/c = 0.5, for every curve were also obtained. The nature of
mechanics (LEFM), and parameters like stress intensity factor relationship between K 1 and f 0.5 , and that for K 2 and f 0.5 were
and fracture energy were applied to model crack propagation examined and then related to f 0.5 through appropriate linear
from an existing crack tip at macro level. At the microstructure fit. The coefficient of correlation in all cases was mostly more
level, fracture behaviour is governed by inherent pores and the than 0.9.
Griffith’s framework holds good for understanding the behaviour.
Thus, strain energy release rate is equated to the rate of surface Micro-structure-based understanding of strength of cement-
energy required for creating new surfaces with respect to pore based materials can be elucidated following the concept
size to obtain the critical tensile stress required for fracture. expressed in Equation 5. An extension of Equation 5 to the
Failure under uniaxial compression is due to tension along a inherently porous concrete with pores ranging over a wide range
direction normal to compression direction, thus compressive of sizes, shapes, and configurations etc., can be represented
strength can be related to pore size parameters, surface energy, approximately using Equation 7 .
[15]
and elastic modulus etc [4,7,12] . Griffith employed the closed form
solution provided by Inglis for stress field around an elliptic hole (7)
in a solid plate and derived the decrease in elastic strain energy
due to crack growth and the rate of decrease with respect to The E 0 and T 0 , represent elastic modulus and surface energy
the half crack length c. The strain energy U from the solution is of permeable-pore free solid, respectively. The apparent solid
U = (πc s )/E; while for 2c crack width and for two new surfaces in this context may, however, contain minor flaws compared
2 2
the total surface energy required S = 4cg; where, g is the surface to micro and meso-pores, hence would undergo deformation
energy per unit area. Equating dU/dc = dS/dc yields the critical under very high-stress, also the surface energy required would
tensile stress at failure as given in Equation 5 [4,7] , be very high. The term (1 – p) represents impermeable solid
fraction with p being the porosity. The term r m represents
(5) median (or log mean) porosity, when volume of permeable
pores is plotted against log apparent pore radius (r). To revisit
where, the parameter c can be considered as representing the Abrams’ Law, it is necessary to relate p and r m to w/c. While
equivalent pore radius in an un-cracked paste or concrete. The relationship between p and w/c for OPC paste is already
equivalent E and g for the material would depend upon porosity. available through Power’s equations (Equations 2 - 4), to obtain a
Thus, the strength of a material like concrete depends upon relationship between r m and w/c, a generalized equation of pore
elastic modulus and surface energy of the skeleton, besides size distribution was proposed as shown in Equation 8. In this
porosity and a representative pore size. Since both porosity equation, the pore radius (r) and corresponding fractional pore
and pore size are functions of the w/c, the influence of w/c on volume (V) are expressed in terms of p, r m , and the dispersion
strength is inherently embedded within this framework. The coefficient d [16-17] . This equation was obtained through regression
equivalent E and g would also depend upon additional factors, analysis of a large data set generated at IIT Delhi, using
including the proportion of un-hydrated cement, aggregate mercury intrusion porosimetry (MIP) and back scattered electron
volume and mineralogy, and on the proportion of hydration microscopy (BSEM). Although BSEM misses the large size pores,
products etc. Thus, while clarity with respect to the effect of w/c the measured distributions were found to follow Equation 8,
is somewhat there, there is considerable ambiguity about other which is based on the Morgan-Mercer-Flodin (MMF) model.

12 THE INDIAN CONCRETE JOURNAL | JANUARY 2026