STRAIN SOFTENING SOLIDS

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STRAIN SOFTENING SOLIDS
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ABSTRACT An analytical investigation for the
propagation of cohesive cracks in a beam of
quasi-brittle material such as concrete is
presented using the fictitious crack model. The
stress-crack opening displacement relation is
assumed as a generalized power law function.
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The analysis gives the size effect and snapback
behavior of beams of softening materials.
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INTRODUCTIONIt is known that a class of
materials such as concrete, rock, brick and
ceramics exhibit strain- softening behavior.
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• The ultimate behavior of such materials is
  characterized by the localization of a non-linear
  zone within a narrow band of the material.

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In this model the strain-softening Behavior is
expressed by a suitable s w relationship. In
general, the relation ship is non-linear. The
fracture energy G is given by the area under
the curve drawn between s w relationship.
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s
Non-linear
G
w
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DESCRIPTION OF THE MODEL. A simply supported beam
in three point bending is considered.The model
assumes that a single fictitious crack develops
in the midsection of the beam when the tensile
stress reaches its ultimate value.
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p
d
t
h
L
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• BASIC ASSUMPTIONS.
• The elastic response of the beam is linear
  outside the cohesive zone.
• A crack is initiated at a point when the maximum
  normal stress reaches the tensile strength.
• After the crack is initiated, the fictitious
  crack progresses.

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(d) When the crack opening displacement reaches a
critical value, the stress transfer becomes zero
and real crack starts to grow. (e) The
compression behavior is assumed linearly elastic.
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The complete flexural behavior is divided into
three phases. Phase I- Before the ultimate
tensile strength is reached in the tensile side
of the beam. Phase II- Development of a
fictitious crack in the elastic layer.
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Phase III-Propagation of real crackand
fictitious crack.
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Stress distribution diagram
Phase-III
Phase-II
Phase-I
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Moment rotation curve
moment
Softening behavior
rotation
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STUDY OF SIZE EFFECT It is clear by this study
that a beam made of strain-softening material
like Concrete exhibits size effects indicating
that the larger size specimens have reduced
strength. From this study it is observed that as
the size-scale increases, the strength reduces.
The rate of decrease of strength reduces as the
value of size-scale increases.
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Also the strength tends to reach a constant value
for large structures. This result is in good
agreement with the studies of Ozbolt et.al. It
may be stated here that the studies of Gerstle
et.al., also show that for concrete flexural
members of very large size the normalized moment
reaches a constant value which is same in this
study also.
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2.5
n1.0
0.5
2.0
0.3
1.5
peak
moment
1.0
0.01
0.001
0.1
1.0
10.0
Size scale
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EFFECT OF BRITLENESS OF THE MATERIAL. The
brittleness is an important parameter in this
study of crack propagation of strain-softening
materials. The curves reveal that for
strain-softening materials the ratio of bending
strength to tensile strength is always greater
than unity.
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It can also be noticed that as the brittleness of
the material increases the strength decreases.It
may be noted that these curves are general in
nature for the fictitious crack model considered
here and results can be used for beam of any
dimension.
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B0.0002
0.002
0.005
moment
0.01
Strain softening exponent
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STUDY OF SNAPBACK BEHAVIOR The present study
indicates that moment- rotation curve will have a
negative slope in the post-peak range. This
indicates that crack formation in concrete beams
leads to unstable behavior in certain cases.
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Snap back effect of post peak
moment
rotation
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CONCLUDING REMARKS. (1)A detailed investigation
on the progressive crack development in a beam of
strain-softening material is presented.This is an
extension of the work by Ulfkjaer et al., who
have considered a linear softening relation.
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n1.0
(linear)
n0.7
n0.5
(non-linear)
n0.3
MOMENT
ROTATION
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(2) The bending strength increases as the
material becomes more soft. (3)It is found that
the snapback effect depends on the brittleness of
the beam. (4) The model is applicable to beams
for which brittleness numbers are less than
1.0 and hence not applicable to large brittle
beams for which linear elastic theory is
applicable.
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(5)In the present model the well-known simple
beam theory is used and hence it is valid for
shallow beams whose slenderness ratio is greater
than 4. (6) The effect of size of the beam is
presented.