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m
2
>m
1
, andexpression(13) canalso be written in another
hardening,than
form(see[21])
02
)
ʳ
f
1
(˙
ʵ
m
1
(
R
1
+
D
1
)
−
m
2
˃
=
(16)
Duetoarbitrarystrainratesatwhichthespecialgranularstructurescanwork,
thelimitofthestrainrate(parameter
D
0
)isassumedas
D
0
=10
4
s
−
1
.
Inthenextstepparametersnand
R
0
arecalculatedfromtheformula
˃
02
=
2
+1
ln
2
−
1
2
n
n
D
0
√
3˙
R
0
(17)
n
ʵ
02
obtainedfrom(10)and(11)forlowvaluesofinelasticstrains(0.2%).Forsuch
small inelastic strains, the isotropic hardening parameter is practically equal
to its initial value (
R
=
R
0
) and the kinematic hardening can be neglected
(
=0). The expression(17)can be treated asthe definition ofthe elasticity
limit.Parameters
D
R
0
arethenidentifiedbytheleast-squaresmethodfrom
theknownvaluesofexperimentallydeterminedproofstressvalues
n
and
˃
02
evaluated
separatelyforeachgroupofexperimentaltests carriedout with variousstrain
rates.
Nextparameters
D
1
and
R
1
canbecalculatedfromthefollowingformulas:
m
2
˃
b
m
1
˃
s
D
1
=
m
2
−
m
1
)
−
m
2
−
m
1
)
−
R
0
(18)
02
)(
02
)(
f
1
(˙
ʵ
f
1
(˙
ʵ
m
2
˃
s
m
2
˃
b
R
1
=
R
0
m
2
−
m
1
)
−
m
2
−
m
1
)
+
(19)
02
)(
02
)(
f
1
(˙
ʵ
f
1
(˙
ʵ
obtainedfrom expressions(15)and(16) for
ʳ
=0,where
˃
b
and
˃
s
areinter-
sectionpointsoflinestangentialto
withtheforceaxis[13].
A similar algorithm has been successfully applied in [18] for evaluation of
Bodner-Partommodelparametersformetals.Thesameuniversalmethodology
has been implemented to identification of the model parameters for polymers
[13],textilefabric”Panama”andsolidpropellants[20].Resultspresentedinthe
nextsectionrevealthatitisalsovalidforinvestigatedspecialgranularstructures.
Manydifferentidentificationmethodshavebeenproposedsofartoevaluate
parameters of various constitutive laws. A short survey of existing ”classical”
approacheshavebeenpresentedin[22].Recently,newoptimizationapproaches
such asgenetic programming,genetic algorithms,and evolutionaryalgorithms
areappliedtosolveinverseproblems[19].However,itneedsto beemphasized
thatapplicationofthenumericalstrategytoestimationofselectedviscoplastic
modelparametersrequiresalotofscientificexperiencefromtheresearcher.
ʳ
4 Verification of Experimental and Numerical Data
According to the analytical methodology of parameter value estimation pre-
sented in Sec. 3, the Bodner-Partom model has been identified for uniaxially
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