Graphical Abstract Figure
Graphical Abstract Figure
Close modal

Abstract

Theoretical and numerical analyses are conducted to rigorously construct master curves that can be used for interpretation of displacement-controlled poroelastic spherical indentation test. A fully coupled poroelastic solution is first derived within the framework of Biot’s theory using the McNamee–Gibson displacement function method. The fully saturated porous medium is assumed to consist of slightly compressible solid and fluid phases and the surface is assumed to be impermeable over the contact area and permeable everywhere else. In contrast to the cases in our previous studies with an either fully permeable or impermeable surface, the mixed drainage condition yields two coupled sets of dual integral equations instead of one in the Laplace transform domain. The theoretical solutions show that for this class of poroelastic spherical indentation problems, relaxation of the normalized indentation force is affected by material properties through weak dependence on a single-derived material constant only. Finite element analysis is then performed in order to examine the differences between the theoretical solution, obtained by imposing the normal displacement over the contact area, and the numerical results where frictionless contact between a rigid sphere and the poroelastic medium is explicitly modeled. A four-parameter elementary function, an approximation of the theoretical solution with its validity supported by the numerical analysis, is proposed as the master curve that can be conveniently used to aid the interpretation of the poroelastic spherical indentation test. Application of the master curve for the ramp-hold loading scenario is also discussed.

References

1.
Oyen
,
M. L.
,
2008
, “
Poroelastic Nanoindentation Responses of Hydrated Bone
,”
J. Mater. Res.
,
23
(
5
), pp.
1307
1314
.
2.
Galli
,
M.
, and
Oyen
,
M. L.
,
2008
, “
Spherical Indentation of a Finite Poroelastic Coating
,”
Appl. Phys. Lett.
,
93
(
3
), p.
031911
.
3.
Hu
,
Y.
,
Zhao
,
X.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2010
, “
Using Indentation to Characterize the Poroelasticity of Gels
,”
Appl. Phys. Lett.
,
96
(
12
), p.
121904
.
4.
Hu
,
Y.
,
Chen
,
X.
,
Whitesides
,
G. M.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2011
, “
Indentation of Polydimethylsiloxane Submerged in Organic Solvents
,”
J. Mater. Res.
,
26
(
6
), pp.
785
795
.
5.
Hu
,
Y.
,
Chan
,
E. P.
,
Vlassak
,
J. J.
, and
Suo
,
Z.
,
2011
, “
Poroelastic Relaxation Indentation of Thin Layers of Gels
,”
J. Appl. Phys.
,
110
, p.
086103
.
6.
Kalcioglu
,
Z. I.
,
Mahmoodian
,
R.
,
Hu
,
Y.
,
Suo
,
Z.
, and
Van Vliet
,
K. J.
,
2012
, “
From Macro-to Microscale Poroelastic Characterization of Polymeric Hydrogels Via Indentation
,”
Soft Matter
,
8
(
12
), pp.
3393
3398
.
7.
Hu
,
Y.
,
You
,
J.-O.
,
Auguste
,
D. T.
,
Suo
,
Z.
, and
Vlassak
,
J. J.
,
2012
, “
Indentation: A Simple, Nondestructive Method for Characterizing the Mechanical and Transport Properties of PH-Sensitive Hydrogels
,”
J. Mater. Res.
,
27
(
1
), pp.
152
160
.
8.
Lai
,
Y.
, and
Hu
,
Y.
,
2017
, “
Unified Solution for Poroelastic Oscillation Indentation on Gels for Spherical, Conical and Cylindrical Indenters
,”
Soft Matter
,
13
(
4
), pp.
852
861
.
9.
Lai
,
Y.
, and
Hu
,
Y.
,
2018
, “
Probing the Swelling-Dependent Mechanical and Transport Properties of Polyacrylamide Hydrogels Through AFM-Based Dynamic Nanoindentation
,”
Soft Matter
,
14
(
14
), pp.
2619
2627
.
10.
Esteki
,
M. H.
,
Alemrajabi
,
A. A.
,
Hall
,
C. M.
,
Sheridan
,
G. K.
,
Azadi
,
M.
, and
Moeendarbary
,
E.
,
2020
, “
A New Framework for Characterization of Poroelastic Materials Using Indentation
,”
Acta Biomater.
,
102
, pp.
138
148
.
11.
Wang
,
M.
,
Liu
,
S.
,
Xu
,
Z.
,
Qu
,
K.
,
Li
,
M.
,
Chen
,
X.
,
Xue
,
Q.
,
Genin
,
G. M.
,
Lu
,
T. J.
, and
Xu
,
F.
,
2020
, “
Characterizing Poroelasticity of Biological Tissues by Spherical Indentation: An Improved Theory for Large Relaxation
,”
J. Mech. Phys. Solids
,
138
, p.
103920
.
12.
Islam
,
M. R.
, and
Oyen
,
M. L.
,
2021
, “
A Poroelastic Master Curve for Time-Dependent and Multiscale Mechanics of Hydrogels
,”
J. Mater. Res.
,
36
(
1
), pp.
2582
2590
.
13.
Greiner
,
A.
,
Reiter
,
N.
,
Paulsen
,
F.
,
Holzapfel
,
G. A.
,
Steinmann
,
P.
,
Comellas
,
E.
, and
Budday
,
S.
,
2021
, “
Poro-Viscoelastic Effects During Biomechanical Testing of Human Brain Tissue
,”
Front. Mech. Eng.
,
7
, p.
708350
.
14.
Faber
,
J.
,
Hinrichsen
,
J.
,
Greiner
,
A.
,
Reiter
,
N.
, and
Budday
,
S.
,
2022
, “
Tissue-Scale Biomechanical Testing of Brain Tissue for the Calibration of Nonlinear Material Models
,”
Curr. Prot.
,
2
(
4
), p.
e381
.
15.
Hertz
,
H.
,
1881
, “
On Contact Between Elastic Bodies
,”
J. für die reine und angewandte Mathematik
,
92
, pp.
156
171
.
16.
Agbezuge
,
L. K.
, and
Deresiewicz
,
H.
,
1974
, “
On the Indentation of a Consolidating Half-Space
,”
Israel J. Tech.
,
12
, pp.
322
338
.
17.
Liu
,
M.
, and
Huang
,
H.
,
2021
, “
Poroelastic Response of Spherical Indentation Into a Half Space With an Impermeable Surface Via Step Displacement
,”
J. Mech. Phys. Solids
,
155
, p.
104546
.
18.
Liu
,
M.
,
2021
, “
On Poroelastic and Poro-Elasto-Plastic Hertzian Contact Problems
,” Ph.D. thesis,
Georgia Institute of Technology
,
Atlanta, GA
.
19.
Mak
,
A. F.
,
Lai
,
W. M.
, and
Mow
,
V. C.
,
1987
, “
Biphasic Indentation of Articular Cartilage I: Theoretical Analysis
,”
J. Biomech.
,
20
(
7
), pp.
703
714
.
20.
Chan
,
E. P.
,
Hu
,
Y.
,
Johnson
,
P. M.
,
Suo
,
Z.
, and
Stafford
,
C. M.
,
2012
, “
Spherical Indentation Testing of Poroelastic Relaxations in Thin Hydrogel Layers
,”
Soft Matter
,
8
(
5
), pp.
1492
1498
.
21.
Liu
,
M.
, and
Huang
,
H.
,
2018
, “
Poroelastic Response of Spherical-Tip Indentation
,”
Proceedings of 52nd US Rock Mechanics/Geomechanics Symposium
,
American Rock Mechanics Association
,
Seattle, WA
.
22.
Liu
,
M.
, and
Huang
,
H.
,
2019a
, “
Poroelastic Response of Spherical Indentation Into a Half Space With a Drained Surface Via Step Displacement
,”
Int. J. Solids Struct
,
165
, pp.
34
49
.
23.
Liu
,
M.
, and
Huang
,
H.
,
2021
, “
Finite Element Modeling of Spherical Indentation in a Poro-Elasto-Plastic Medium Via Step Displacement Loading
,”
Int. J. Num. Anal. Meth. Geomech.
,
45
(
10
), pp.
1347
1380
.
24.
Liu
,
M.
, and
Huang
,
H.
,
2023
, “
Legitimacy of the Hertzian Assumptions for Poroelastic Spherical Indentation
,”
Proceedings of 57th US Rock Mechanics/Geomechanics Symposium
,
Atlanta, GA
,
June 25–28
,
American Rock Mechanics Association
.
25.
Love
,
A. E. H.
,
1929
, “
The Stress Produced in a Semi-Infinite Solid by Pressure on Part of the Boundary
,”
Philos. Trans. R. Soc. A
,
228
(
659–669
), pp.
377
420
.
26.
Liu
,
M.
, and
Huang
,
H.
,
2016
, “
Sphere Indentation – The Hertzian Stress Field and the Effect of Far-Field Confining Stress
,” Proceedings of 50th US Rock Mechanics/Geomechanics Symposium,
American Rock Mechanics Association
,
Houston, TX
.
27.
McNamee
,
J.
, and
Gibson
,
R. E.
,
1960
, “
Displacement Functions and Linear Transforms Applied to Diffusion Through Porous Elastic Media
,”
Q. J. Mech. Appl. Math.
,
13
(
1
), pp.
98
111
.
28.
Verruijt
,
A.
,
1971
, “
Displacement Functions in the Theory of Consolidation Or in Thermoelasticity
,”
Zeitschrift für angewandte Mathematik und Physik ZAMP
,
22
(
5
), pp.
891
898
.
29.
Verruijt
,
A.
,
2013
, “
Theory and Problems of Poroelasticity
,”
Delft University of Technology
.
30.
Noble
,
B.
,
1963
, “
The Solution of Bessel Function Dual Integral Equations by a Multiplying-Factor Method
,”
Math. Proc. Camb.
,
59
(
2
), pp.
351
362
.
31.
Zemyan
,
S. M.
,
2012
,
The Classical Theory of Integral Equations: A Concise Treatment
,
Springer
.
32.
Wynn
,
P.
,
1956
, “
On a Device for Computing the em (Sn) Transformation
,”
Math. Tab. Aids Comp.
,
10
(
54
), pp.
91
96
.
33.
Bracewell
,
R. N.
,
1986
,
The Fourier Transform and Its Applications
,
McGraw-Hill
,
New York
.
34.
Poularikas
,
A. D.
,
2018
,
Transforms and Applications Handbook
,
CRC Press
,
Boca Raton, FL
.
35.
Stehfest
,
H.
,
1970
, “
Algorithm 368: Numerical Inversion of Laplace Transforms
,”
Commun. ACM
,
13
(
1
), pp.
47
49
.
36.
Cheng
,
A. H.-D.
,
2016
,
Poroelasticity
,
Springer
,
New York
.
37.
Hay
,
J. L.
, and
Wolff
,
P. J.
,
2001
, “
Small Correction Required When Applying the Hertzian Contact Model to Instrumented Indentation Data
,”
J. Mater. Res.
,
16
(
5
), pp.
1280
1286
.
38.
Wriggers
,
P.
,
2006
,
Computational Contact Mechanics
,
Springer
,
New York
.
39.
Wriggers
,
P.
, and
Imhof
,
M.
,
1993
, “
On the Treatment of Nonlinear Unilateral Contact Problems
,”
Arch. Appl. Mech.
,
63
(
2
), pp.
116
129
.
You do not currently have access to this content.