Modelling of Innovative Controllable Structures Made of Granular Materials - New Contributions in Information Systems and Technologies - page 264

the function's character is small. Also, the isotropic and kinematic hardening
functionsaredefinedforawholerangeofdeformation.
Initsgeneralform,theBodner-Partomconstitutivemodelhastwelvemate-
rialparameters.Whenthedeformationprocessisrapid,theinfluenceofrecov-
ery can be neglected, and the number of parameterscan be reduced to seven:
n,D 0 ,D 1 ,R 0 ,R 1 ,m 1 ,m 2 .
Letusintroducethefunctional

f linkingthespecificloadfunction ˃ withthe

I

functionoftheinelasticstrainrate ˙ ʵ

˃

I )

= f (˙ ʵ (10)

R + D

Thefunctional f 1 intheBodner-Partommodelhastheform:

f 1 = 2

+1 ln 2

2 n

n D 0

√ 3˙

(11)

n

I

ʵ

Asinthefirststepappliedinthepaperidentificationmethodology,functions
of the strain

I
necessaryintheidentificationprocesshavebeencalculatedonthebasisofthe
experiments reported in Sec. 2, and then initially pretreated according to re-
marksin[13].Thenextstageiseliminatingtheinfluenceofexperimentalerror.
To provideaccuratedata, the graphofthe inelastic strainrate-stressfunction
hasbeen approximatedbythe multiparameterexponentialfunction[12]. Cal-
culatingderivativeofthepreviouslymentionedapproximationfunctionenables
establishingthehardeningworkratefunction

I and inelastic strain rate ˙

ʵ , specific load ˃ , inelastic strain ʵ ʵ

ʳ = d˃
dW

(12)

I

andaccordingto(3)and(10)itcanbeexpressedas

ʳ = f 1 [ m 1 ( R 1 − R )+ m 2 ( D 1 − D )] (13)

Afterward, the graph of the function has to be plotted to estimate values
of parameters m1 and m2. The value of the parameter m2 is estimated for
very small inelastic strains (less than 0.2%) according to (10) which can be
writtenas:

= ˃
f 1 (˙

˃
f 1 (˙

D

02 ) − R ≈ 02 ) − R 0

(14)

ʵ ʵ

From(13)theaboveexpressionisgivenby

02 )[

ʳ = f 1 (˙ ʵ m 1 ( R 1 − R 0 )+ m 2 ( R 0 − D 1 )] − m 2 ˃ (15)

m 1 can be determined in the upper range of small in-
elastic strains (less than 1%). In such cases it is assumed that the hardening
functions areclose to maximum values (saturationof the hardening). Because
the kinematichardening function generallysaturatesearlierthan the isotropic

Now the parameter