FINER-SCALE EXTRACTION OF VISCOELASTIC PROPERTIES FROM
MATERIALS EXHIBITING ELASTIC, VISCOUS, AND PLASTIC MATERIAL
BEHAVIOR
A. Jäger1, R. Lackner1,2, and Ch. Pichler1
Institute for Mechanics of Materials and Structures, Vienna University of Technology, Vienna,
Karlsplatz 13/202, A-1040
2
Computational Mechanics, Technical University of Munich, Munich, Arcisstraße 21, D-80333
1
ABSTRACT
Motivated by recent progress in viscoelastic indentation analysis, the identification of viscoelastic properties from materials
exhibiting elastic, viscous, and plastic material behavior by means of nanoindentation is dealt with in this paper. Based on
existing solutions for pure viscoelastic material behavior, the so-called double indentation technique is presented. This method
is applied to three different polymers, giving access to the model parameters of the fractional dashpot used to describe the
viscoelastic behavior. The obtained results are compared with results from standard (single) indentation tests using a
Berkovich indenter. Moreover, the influence of the maximum load, determining the amount of plastic material response, on the
identified model parameters was investigated. Finally, the creep compliance obtained from the preformed indentation tests is
compared with values for the elastic deviatoric compliance reported in the literature.
Introduction – single- versus double-indentation
Recent progress in both finer-scale experimentation (atomic force microscopy, nanoindentation, ...) and theoretical and
numerical upscaling schemes provides the basis for the development of so-called multiscale models, taking finer scales of
observation into account. The success of multiscale models is strongly linked to the proper identification of material properties
at finer scales, serving as input for the mentioned upscaling schemes. In this paper, extraction of viscoelastic material
parameters by means of the nanoindentation technique is presented. Parameter identification of materials exhibiting only
elastic and time-dependent material response is based on the measured increase of penetration during the so-called holding
phase of the measured penetration history [3-5] (see Figure 1(a)). Hereby, the experimental data are compared with the
respective analytical solution for the mathematical problem of a rigid indenter penetrating a viscoelastic medium, giving access
to the parameters of the underlying viscoelastic model. Such back-analysis schemes, however, are only applicable to materials
showing no plastic behavior. In case of plastic deformations, occurring during the loading phase of the NI test, both plastic and
viscoelastic deformations would be assigned to the viscoelastic material response, leading to wrong parameters for the
underlying viscoelastic model.
Recently, pre-loading of the material was performed in [7], leading to the so-called double-indentation technique (see
Figure 1(b)). Hereby, the load history is characterized by two load cycles, with the first cycle (pre-loading) leading to plastic
deformations and the second cycle showing only viscoelastic deformations. Thus, the latter is suitable for back-calculation of
viscoelastic properties using analytical solutions for the viscoelastic indentation problem. The required analytical solution for
the penetration of a tip into the imprint resulting from the first load cycle is based on the concept of “effective indenters”
proposed by Sakai [8]. According to Sakai, the reloading indentation process (second cycle) into a plastic imprint (stemming
from the first cycle) can be approximated by the indentation of a tip with an effective tip shape into a flat undeformed surface.
Hereby, the semi-apex angle β of the original indenter is replaced by an effective semi-apex angle β eff = β − β r ,
with tan β r = hr / hc tan β . Hereby, hr is the residual penetration after the first load cycle and hc is the contact height at
maximum load of the first load cycle. Following this idea, adaptation of the tip shape f ( ρ ) already used in the identification
scheme reported in [5] is proposed in this paper, giving the effective tip shape as
f eff ( ρ ) = (1 − hr / hmax ) f ( ρ ) .
(1)
The residual penetration hr and the maximum penetration hmax are set equal to the penetrations at points 4 and 2,
respectively (see Figure 1(b)). Point 4 is chosen in order to account for viscoelastic deformations during the holding period
after unloading, reducing the residual penetration depth. Due to the viscoelastic deformation occurring between points 2 and 3,
the initial slope of the unloading curve, required for the determination of the contact height hc , may become negative. Hence,
hc was replaced by the maximum penetration hmax for the determination of f eff ( ρ ) in Equation (1). The identification of model
parameters is identical to the identification scheme proposed in [5], considering the holding phase of the second load cycle
and the effective tip shape f eff ( ρ ) .
Figure 1. Load-penetration curve, load history, and penetration history for (a) single-indentation test and (b) double-indentation
test on material exhibiting elastic, viscous, and plastic material response
Methods – identification of viscoelastic properties
For identification of viscoelastic properties, the procedure outlined in [5] is followed. Hereby, the tip shape is approximated
by
Atip = ρ 2 π = C 0 f ( ρ ) 2 + C1 f ( ρ ) ,
(2)
where Atip is the cross section area of the tip, and
f (ρ ) =
1 ⎛
2
2
⎞
⎜ C1 + 4C 0 ρ π − C1 ⎟ [m]
⎠
2C 0 ⎝
(3)
is the distance from the apex of the axisymmetric tip as a function of the radius ρ [m]. C 0 [-] and C1 [m] are constants
describing the tip shape, obtained from calibration of the NI-testing equipment.
Based on the analytical solution of the elastic indentation problem [2] for the tip shape given in Equation (3), the
viscoelastic solution is obtained by the application of the method of functional equations [9], giving [5]
4a 2 C 0 π
dP (τ )
2a 3
) = Y (t − τ )
dτ .
2 F1 (0.5;2;2.5;−
2
dτ
3C1
C1
0
t
4π
∫
(4)
Hereby, the left-hand side of Equation (4) depends only on geometric properties, i.e., the parameters C 0 and C1 , and on the
experimentally-obtained contact radius a (t ) . The right-hand side of Equation (4), on the other hand, depends on the
mechanical properties of the viscoelastic halfspace represented by the so-called indentation compliance function Y (t ) and the
load history P (t ) . In Equation (4), 2 F1 ( a; b; c; z ) denotes a hypergeometric function, and the unknown contact radius a is
determined from the measured penetration depth h , using
h = a π / C 0 arctan(2a C 0 π / C1 ) .
(5)
For identification of viscoelastic model parameters, a single non-linear (fractional) dash-pot is used. The respective creepcompliance function reads
J nlDP = J a (t / τ ) k ,
(6)
-1
where J a [GPa ] is the initial creep compliance and k [-] is the creep exponent. τ is introduced for dimensional reasons and
is set equal to 1s. The fractional dash-pot is able to cover a wide range of material behavior from pure elastic ( k =0) to pure
viscous ( k =1) materials. Considering the indentation compliance function for incompressible materials, with Y (t ) = 1 / 4 J , the
right-hand side of Equation (4) reads for the fractional dash-pot and the holding phase of a trapezoidal load history [6]
t
∫
FH − nlDP (t ) := 1 / 4 J nlDP (t − τ )
0
P
dP (τ )
dτ = max
dτ
4τ L
k
⎧⎪
1 ⎛ 1 ⎞ k +1
k +1
⎜ ⎟ t − (t − τ L )
⎨J a
⎪⎩ k + 1 ⎝ τ ⎠
[
⎫
]⎪⎬ . (7)
⎪⎭
In order to determine model parameters from NI-test data, the error between the left-hand side of Equation (4), denoted as
Fexp ( a) , and FH − nlDP (t ) given in Equation (7) is minimized within the holding phase by adapting the unknown model parameters
J a and k . The mentioned error is defined by
R nlDP ( J a , k ) =
e( J a , k )
.
u
(8)
with
e2 (J a , k) =
∑ (F
n
exp
{a[h(t i )]} − FH − nlDP ( J a , k , t i )
)
2
and u 2 =
i =1
∑ (F
n
exp
)
2
(t i ) ,
(9)
i =1
with t i denoting different time instants within the holding phase.
As regards the double-indentation technique, the tip-shape function f ( ρ ) is replaced by the effective tip shape
f eff ( ρ ) = (1 − hr / hmax ) f ( ρ ) . Considering f eff ( ρ ) in the identification scheme, only the functions describing the tip shape given
in the left-hand side of Equation (4) and Equation (5) are adapted, reading
Fexp,eff ( a) = (1 − hr / hmax ) Fexp ( a)
and
heff = (1 − hr / hmax ) h .
Materials and experimental setup
Three different polymers, i.e., low density polyethylene (LDPE), polymethyl methacrylate (PMMA), and polycarbonate (PC),
are considered in this study. The respective values for Young’s modulus and Poisson’s ratio were taken from the literature and
are listed in Table 1.
Table 1. Young’s modulus and Poisson’s ratio of considered polymers
polymer
(source)
LDPE
(www.matweb.com)
PMMA
(www.matweb.com)
PC
(Bayer Industry Services)
Young’s modulus [GPa]
0.139-0.35
1.8-3.1
2.3-2.4
Poisson’s ratio [-]
0.38
0.35-0.4
0.37
NI tests were conducted at room temperature using a Hysitron Triboindenter nanoindenter with a three-sided Berkovich
diamond tip. The area function of the Berkovich tip was obtained by the standard calibration procedure outlined in [1]. Figure 2
shows the tip-shape function f ( ρ ) and the area function Atip ( f ( ρ )) for the Berkovich tip.
Figure 2. (a) Tip-shape function f ( ρ ) and (b) area function Atip ( f ( ρ )) for the Berkovich tip ( C 0 =24.5, C1 =2314 nm)
The load history of the single indents (see Figure 1(a)) is characterized by a loading phase with a duration of 3 or 6s followed
by a holding phase of 20s at maximum load. During double indentation, the load was applied in 0.2s in the first load cycle and
then reduced to 1 μN in 0.2s. After a holding phase of 20s at a load of 1μN, the load was increased again in 3 or 6s to 50% of
the load considered in the first load cycle, and held constant for 20s.
Results and discussion
All three polymers were tested at different maximum loads using the single- and double-indentation method. For each
maximum load, at least 9 indents are conducted and mean values for the identified parameters for the fractional dashpot, J a
and k , are computed. Figures 3-5 show the load dependence of the initial creep compliance J a and the creep exponent k
for LDPE, PMMA, and PC obtained for single- and double-indentation. For both methods, a similar load dependence is
obtained for all three materials, reaching a constant value of both parameters as the maximum load increases (see Table 2).
The variation of the identified model parameters at small loads is assigned to the influence of the roughness of the sample
surface which is more pronounced for small penetrations. The different loading times considered in the NI tests (3 and 6s)
have only marginal impact on the identified model parameters. Whereas the same initial creep compliance is obtained for the
two loading times, the creep exponent is slightly higher for the loading time of 6s.
Comparing the identified parameters obtained from single- and double-indentation, the single-indentation was found to
overestimate the initial creep compliance for all three polymers. This difference in the identified parameters is associated with
the occurrence of plastic deformations. The creep exponent, on the other hand, is only slightly underestimated by the singleindentation method.
Table 2. Initial creep compliance J a and creep exponent k for considered polymers obtained from single- and doubleindentation testing
Initial creep compliance
Test method
LDPE
PMMA
PC
single-indentation
12.28
1.06
1.82
J a [GPa ]
double-indentation
9.55
0.80
1.01
Creep exponent
single-indentation
0.098
0.093
0.036
k [-]
double-indentation
0.103
0.105
0.050
-1
Figure 3. Fractional dash-pot parameters for LDPE obtained from (a) single- and (b) double-indentation for
different values of Pmax
Figure 4. Fractional dash-pot parameters for PMMA obtained from (a) single- and (b) double-indentation
for different values of Pmax
Figure 5. Fractional dash-pot parameters for PC obtained from (a) single- and (b) double-indentation
for different values of Pmax
In order to compare the obtained model parameters with material properties obtained from macroscopic testing, the creep
compliance (Equation (6)) is plotted for all three polymers using the parameters given in Table 2 (see Figure 6) and compared
with the elastic deviatoric compliance determined from the macroscopic material parameters given in Table 1, using
G = E /(2(1 + ν )) . In contrast to the results obtained from single-indentation, the creep compliance corresponding to model
parameters identified via double-indentation lies within the range of the elastic compliance for all three considered materials.
Especially in case of PC, showing the highest amount of plastic deformations, the development of plastic deformations during
the loading phase lead to a significant overestimation of the creep compliance based on the parameters identified via singleindentation. The creep compliance computed from the model parameters from double-indentation, on the other hand, agrees
perfectly with the elastic compliance from the literature.
Figure 6. Creep compliance obtained from single- and double-indentation for (a) LDPE, (b) PMMA, and (c) PC compared with
elastic compliance given in Table 1
Summary and conclusions
The identification of viscoelastic model parameters from indentation tests was dealt with in this paper. Accounting for the
plastic material response during the loading phase of indentation tests, results from double-indentation applied to three
different polymers were presented. The parameters for the fractional dash-pot, describing the viscoelastic behavior of the
considered polymers, were determined from back-calculation using the increase of penetration during the holding phase of the
indentation test. Based on the so-obtained model parameters, the following conclusions can be drawn:
1.
2.
3.
For both single- and double-indentation the model parameters became constant for an increasing maximum load. The
small variation of the identified model parameters for small loads was assigned to the roughness of the sample
surface.
The initial creep compliance of the fractional dash-pot identified via double-indentation was lower than the compliance
corresponding to single-indentation. As regards the latter, plastic deformations were assigned to the viscoelastic
material response, explaining the higher values for the initial creep compliance.
The creep compliance obtained from the model parameters identified via double-indentation agreed well with the
elastic deviatoric compliance reported in the literature.
In summary, the double-indentation method was found to be an appropriate tool for considering plastic deformations during
the identification of viscoelastic model parameters for materials exhibiting viscoelastic and plastic material behavior. However,
the conventional single-indentation method may still be applied for materials dominated by viscoelastic material response,
such as e.g. low-density polyethylene.
Acknowledgments
The authors thank the remaining members of the Christian Doppler laboratory for “Performance-Based Optimization of Flexible
Pavements” at Vienna University of Technology for helpful comments and fruitful discussions on the presented research work.
Financial support by the Christian Doppler Gesellschaft (Vienna, Austria) is gratefully acknowledged.
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