International Journal of Fracture, 62 (1993), 355-373.
R-CURVE MODELING OF RATE AND SIZE EFFECTS
Zdenek P. Bazant and Milan Jirásek
IN QUASIBRITTLE FRACTURE
Evanston, Illinois 60208, U.S.A.
The equivalent linear elastic fracture model based on an R-curve
(a curve characterizing the variation of the critical energy release rate
with the crack propagation length) is generalized to describe both the rate
effect and size effect observed in concrete, rock or other quasibrittle
materials. It is assumed that the crack propagation velocity depends on
the ratio of the stress intensity factor to its critical value based
on the R-curve and
that this dependence has the form of a power
function with an exponent much larger than 1. The shape of the
R-curve is determined as
the envelope of the fracture equilibrium curves corresponding to the maximum
load values for geometrically similar specimens of different sizes. The creep
in the bulk of a concrete specimen must be taken into account, which is done
by replacing the elastic constants in the linear elastic fracture mechanics
(LEFM) formulas with a linear viscoelastic operator in time (for rocks, which
do not creep, this is ommited). The experimental observation that
the brittleness of concrete increases as the loading rate decreases
(i.e., the response shifts in the size effect plot closer to LEFM)
can be approximately described by assuming that stress relaxation causes
the effective process zone length
in the R-curve expression to decrease with a decreasing loading rate.
Another power function is used to describe this. Good fits of test data
for which the times
to peak range from 1 sec to 250000 sec are demonstrated. Furthermore,
the theory also describes the recently conducted relaxation tests, as
well as the recently observed response to a sudden change of loading rate
(both increase and decrease), and particularly the fact that a sufficient
rate increase in the post-peak range can produce a load-displacement
response of positive slope leading to a second peak.
Summary and Conclusions
The equivalent linear elastic fracture model based on an $R$-curve
(a curve characterizing the variation of critical energy release rate
with crack propagation length) can be generalized to the rate effect
if the crack propagation velocity is assumed to depend either on the ratio
of the stress intensity factor to its critical value based on the $R$-curve,
or on the difference between these two variables. This dependence may be
assumed in the form of an increasing power function with a large exponent.
The creep in the bulk of a concrete specimen must also be taken into
account, which can be done by replacing the elastic constants in the
LEFM formulas with a linear viscoelastic operator in time. For rocks,
which do not creep, this is not necessary.
The experimental observation that the brittleness of concrete increases
with a decreasing loading rate (i.e., the response shifts in the size effect
plot closer to linear elastic fracture mechanics) can be at least
approximately modeled by assuming the effective fracture process zone length
in the $R$-curve expression to decrease with a decreasing rate. This dependence
may again be described by a power function.
Good agreement with the previous test results for concrete and limestone,
recently measured at very different loading rates, with times to peak
ranging from 1 second to 250000 seconds,
The model can also predict the following phenomena recently observed
in the laboratory: (a) When the loading rate is suddenly increased,
the slope of the load-displacement diagram suddenly increases.
For a sufficient rate increase, the slope becomes positive even
in the post-peak range, and later in the test a second peak,
lower or higher than the first peak, is observed.
(b) When the rate suddenly decreases, the slope suddenly decreases
and the response approaches the load-displacement curve for the lower
rate. (c) When the displacement is arrested, relaxation causes a drop
of load, approximately following a logarithmic time curve.
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