Page 311 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Fracture Mechanisms in Nonmetals 291

6.3 CONCRETE AND ROCK

Although concrete and rock are often considered brittle, they are actually quasi-brittle materials
that are tougher than most of the so-called advanced ceramics. In fact, much of the research on
toughening mechanisms in ceramics is aimed at trying to make ceramic composites behave more
like concrete.
Concrete and rock derive their toughness from subcritical cracking that precedes ultimate
failure. This subcritical damage results in a nonlinear stress-strain response and R-curve behavior.
A traditional strength-of-materials approach to designing with concrete has proved inadequate
because the fracture strength is often size dependent [50]. This size dependence is due to the fact
that nonlinear deformation in these materials is caused by subcritical cracking rather than plasticity.
Initial attempts to apply fracture mechanics to concrete were unsuccessful because these early
approaches were based on linear elastic fracture mechanics (LEFM) and failed to take account of
the process zones that form in front of macroscopic cracks.
This section gives a brief overview of the mechanisms and models of fracture in concrete and
rock. Although most of the experimental and analytical work has been directed at concrete as
opposed to rock, due to the obvious technological importance of the former, rock and concrete
behave in a similar manner. The remainder of this section will refer primarily to concrete, with the
implicit understanding that most observations and models also apply to geologic materials.
Figure 6.39 schematically illustrates the formation of a fracture process zone in concrete,
together with two idealizations of the process zone. Microcracks form ahead of a macroscopic
crack, which consists of a bridged zone directly behind the tip and a traction-free zone further
behind the tip. The bridging is a result of the weak interface between the aggregates and the matrix.
Recall Section 6.2.4, where it was stated that fiber bridging, which occurs when the fiber-matrix
bonds are weak, is the most effective toughening mechanism in ceramic composites. The process
zone can be modeled as a region of strain softening (Figure 6.40(b)) or as a longer crack that is
subject to closure tractions (Figure 6.40(c)). The latter is a slight modification to the Dugdale-
Barenblatt strip-yield model.
Figure 6.40 illustrates the typical tensile response of concrete. After a small degree of nonlin-
earity caused by microcracking, the material reaches its tensile strength σ and then strain softens.
t
Once σ is reached, the subsequent damage is concentrated in a local fracture zone. Virtually all
t
of the displacement following the maximum stress is due to the damage zone. Note that Figure 6.40
shows a schematic stress-displacement curve rather than a stress-strain curve. The latter is inap-
propriate because nominal strain measured over the entire specimen is a function of gage length.
There are a number of models for fracture in concrete, but the one that is most widely referenced
is the so-called fictitious crack model of Hillerborg [51, 52]. This model, which has also been called
a cohesive zone model, is merely an application of the Dugdale-Barenblatt approach. The Hillerborg
model assumes that the stress displacement behavior (σ-δ ) observed in the damage zone of a tensile
specimen is a material property. Figure 6.41(a) shows a schematic stress-displacement curve, and
6.41(b) illustrates the idealization of the damage zone ahead of a growing crack.
At the tip of the traction-free crack, the damage zone reaches a critical displacement δ . The
c
tractions are zero at this point, but are equal to the tensile strength σ , at the tip of the damage zone
t
(Figure 6.39(c)). Assuming that the closure stress σ and opening displacement δ are uniquely
related, the critical energy release rate for crack growth is given by

d
G = ∫ c δ σδ (6.25)
c
0
which is virtually identical to Equation (3.43) and Equation (6.22).
The key assumption of the Hillerborg model that the σ-δ relationship is a unique material
property is not strictly correct in most cases because process zones produced during the fracture