A dynamic integro-differential operator of variable order is suggested for a more adequate description of processes, which involve state dependent measures of elastic and inelastic material features. For any negative constant order this operator coincides with the well-known operator of fractional integration. The suggested operator is especially effective in cases with strong dependence of the behavior of the material on its present state—i.e., with pronounced nonlinearity. Its efficiency is demonstrated for cases of viscoelastic and elastoplastic spherical indentation into such materials (aluminum, vinyl) and into an elastic material (steel) used as a reference. Peculiarities in the behavior of the order function are observed in these applications, demonstrating the “physicality” of this function which characterizes the material state. Mathematical generalization of the fractional-order integration-differentiation in the sense of variability of the operator order, as well as definitions and techniques, are discussed. [S0021-8936(00)02102-4]

Issue Section:
Technical Papers
Keywords:
mechanical contact, integro-differential equations, elasticity, plasticity, viscoelasticity, elastoplasticity, indentation, deformation
Topics:
Deformation, Stress, Aluminum
1.
Billington, E. W., and Tate, A., 1981, The Physics of Deformation and Flow, McGraw-Hill, New York.
2.
Krajcinovic, D., 1996, Damage Mechanics. Elsevier, Amsterdam.
3.
Lee
,
E. N.
, and
Radok
,
J. R. M.
,
1960
, “
The Contact Problem for Viscoelastic Bodies
,”
ASME J. Appl. Mech.
,
82
, pp.
438
–
444
.
4.
Sneddon, J. N., 1979, The Use of Operators of Fractional Integration in Applied Mathematics, PWN, Warsaw.
5.
Bagley
,
R. L.
, and
Calico
,
R. A.
,
1991
, “
Fractional Order State Equations for the Control of Viscoelastically Damped Structures.
”
J. Guidance Control Dynam.
,
14
, No.
2
, pp.
304
–
311
.
6.
Bagley
,
R. L.
,
Swinney
,
D. V.
, and
Griffin
,
K. E.
,
1993
, “
Fractional Order Calculus Model of the Generalized Theodorsen Function
,”
J. Aircr.
,
30
, No.
6
, pp.
1003
–
1005
.
7.
Rossikhin
,
Yu. A.
, and
Shitikova
,
M. V.
,
1997
, “
Application of Fractional Operators to the Analysis of Damped Vibrations of Viscoelastic Single-Mass Systems
,”
J. Sound Vib.
,
199
, No.
4
, pp.
567
–
586
.
8.
Lee
,
E. N.
, and
Rogers
,
T. G.
,
1963
, “
Solution of Viscoelastic Stress Analysis Problems Using Measured Creep or Relaxation Functions
,”
ASME J. Appl. Mech.
,
30
, pp.
127
–
133
.
9.
Rabotnov, Yu. N., 1977, Elements of Hereditary Mechanics of Solids, Nauka, Moscow.
10.
Bardenhagen
,
S. G.
,
Stout
,
M. G.
, and
Gray
,
G. T.
,
1997
, “
Three-Dimensional, Finite Deformation, Viscoplastic Constitutive Models for Polymeric Materials
,”
Mech. Mater.
,
25
, No.
4
, pp.
235
–
253
.
11.
Lubliner, J., 1990, Plasticity Theory, Macmillan, New York.
12.
Skrzypek, J. J., and Hetnarski, R., 1993, Plasticity and Creep, CRC Press, Boca Raton, FL.
13.
Valanis
,
K. S.
,
1971
, “
A Theory of Viscoplasticity Without a Yield Surface I—II
,”
Archiwum Mechaniki Stosowanej
,
23
, No.
4
, pp.
517
–
551
.
14.
Valanis
,
K. C.
, and
Read
,
H. E.
,
1986
, “
An Endochronic Plasticity Theory for Concrete
,”
Mech. Mater.
,
5
, No.
3
, pp.
277
–
295
.
15.
Valanis
,
K. C.
, and
Yilmazer
,
U.
,
1978
, “
An Intrinsic Time to Fracture Criterion for Amorphous Polymers
,”
Eng. Fract. Mech.
,
10
, No.
3
, pp.
659
–
676
.
16.
Rajic
,
N.
, et al.,
1997
, “
Plastic Zone Size Determination by Temperature Measurement
,”
ASME J. Eng. Mater. Technol.
,
119
, pp.
32
–
39
.
17.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
18.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
19.
Sheivekhman
,
A.
, and
Suzdalnitsky
,
J.
,
1997
, “
Determination of the Elastoplastic Properties of Engineering Materials From Spherical Indentation
,”
J. Mech. Behav. Mater.
,
8
, No.
4
, pp.
283
–
294
.
20.
Francis
,
H. A.
,
1976
, “
Phenomenological Analysis of Plastic Spherical Indentation
,”
ASME J. Eng. Mater. Technol.
,
98
, pp.
272
–
281
.
21.
Field
,
J. S.
, and
Swain
,
M. V.
,
1993
, “
A Simple Predictive Model for Spherical Indentation
,”
J. Mater. Res.
,
8
, No.
2
, pp.
297
–
305
.
22.
Field
,
J. S.
, and
Swain
,
M. V.
,
1995
, “
Determining the Mechanical Properties of Small Volumes of Material From Submicrometer Spherical Indentations
,”
J. Mater. Res.
,
10
, No.
1
, pp.
101
–
112
.
23.
Barlow
,
D. A.
,
1973
, “
The Indentation and Scratch Hardness of Plastics
,”
ASME J. Eng. Mater. Technol.
,
95
, pp.
243
–
251
.
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