Civil Engineering Reference
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for the analysis of reinforced concrete structures, and can be used with rebar
to model concrete reinforcement. In addition, the model is designed
for applications in which concrete is subjected to monotonic, cyclic, and/
or dynamic loading under low confining pressures. The model consists of
the combination of nonassociated multihardening plasticity and scalar
(isotropic) damaged elasticity to describe the irreversible damage that occurs
during the fracturing process. Furthermore, the model allows user control of
stiffness recovery effects during cyclic load reversals and can be defined to be
sensitive to the rate of straining. The model can be used in conjunction with
a viscoplastic regularization of the constitutive equations in ABAQUS to
improve the convergence rate in the softening regime.
Concrete damaged plasticity model requires that the elastic behavior of
the material be isotropic and linear. The model is a continuum, plasticity-
based, damage model for concrete. It assumes that the main two failure
mechanisms are tensile cracking and compressive crushing of the concrete
material. The evolution of the yield (or failure) surface is controlled by two
hardening variables,
e
t
pl
and
e
pl
, linked to failure mechanisms under tension
and compression loading, respectively. ABAQUS refers to
e
t
pl
and
e
pl
as
tensile and compressive equivalent plastic strains, respectively. The model
assumes that the uniaxial tensile and compressive response of concrete is
characterized by damaged plasticity, as shown in
Figure 5.24
.
Under uni-
axial tension, the stress-strain response follows a linear elastic relationship
until the value of the failure stress,
s
to
, is reached. The failure stress cor-
responds to the onset of microcracking in the concrete material. Beyond
the failure stress, the formation of microcracks is represented macroscop-
ically with a softening stress-strain response, which induces strain localiza-
tion in the concrete structure. Under uniaxial compression, the response is
linear until the value of initial yield,
s
co
. In the plastic regime, the response
is typically characterized by stress hardening followed by strain softening
beyondtheultimatestress,
s
cu
. This representation, although somewhat
simplified, captures the main features of the response of concrete. It is
assumed that the uniaxial stress-strain curves can be converted into stress
versus plastic-strain curves. As shown in
Figure 5.25
, when the concrete
specimen is unloaded from any point on the strain-softening branch of
the stress-strain curves, the unloading response is weakened: the elastic
stiffness of the material appears to be damaged (or degraded). The degra-
dation of the elastic stiffness is characterized by two damage variables,
d
t
and
d
c
, which are assumed to be functions of the plastic strains, tempera-
ture, and field variables.
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