embedded in an elastic substrate: Theoretical framework and experiment Guoxin
, 
, Zhi-Hui
We propose to measure the elastoplastic properties of micro- and 
nano-fibers by a normal indentation technique in which the vertically aligned fibers
are embedded in an elastic matrix. Measurements are taken at two
different indentation depths, which represent different levels of the
matrix effects and lead to the establishment of two independent
equations that correlate the fiber/matrix
properties with the indentation responses. Effective reverse analysis
algorithms are proposed, and by following which the desired fiber
properties can be determined from a sharp indentation test.
Comprehensive analysis is also carried out to verify the effectiveness
and error sensitivity of the presented method. The extracted material
properties agree well with those measured from the parallel experiments
on human hair and glass fibers.
Keywords: A. Fibres; D. Nanoindentation; B. Mechanical properties; C. Computational modeling
fibernano-fibers have been studied intensively due to their wide applications in tissue engineering [1], biomaterial [2], filter media [3], reinforced composites [4], and micro/nano-electro-mechanical systems [5]. The fibers can be either natural or synthesized: examples of the natural fibers include plant fiber, lignocellulosic fiber, and hair, and manmade fibers
can be synthesized from polymers, carbon, glass, and metal, which can
be either homogeneous or in the form of an annular “layered” coaxial
composite column.In practical applications, fibers
experience various mechanical loads that may cause permanent
deformation and/or malfunction due to failure; therefore,
characterizing the mechanical properties of fibers is the first step of evaluating their structural integrity. For natural fibers,
such as human hair, its mechanical properties can provide useful
information to diagnose hair disorders and to evaluate the response to
therapeutic regimens [6],
thus important for biomedical purposes. Moreover, the variation of
hair’s mechanical properties can help to evaluate the effect of
different cosmetic products (such as shampoo and conditioner), which
has important industrial applications [7] and [8]. Therefore, it is very important to characterize the mechanical properties of a single fiber – although the present study is motivated by the importance of mechanical properties of human hair, the fibers studied in this paper are rather broad and in general can be applied to polymer, glass, ceramic, or high-strength metal fibers.
Conventionally, the mechanical properties of fibers can be measured by tensile test [9] and [10], bending tests [11], or resonant frequency [12]; however, handling and manipulating micro- and nano-fibers are usually difficult and time consuming [13].
Among alternative techniques, nanoindentation is perhaps the most
convenient, simplest, and fastest way of measuring the mechanical
properties of material at small scale [14] and [15]. Since nanoindentation is a compressive test, the fiber must be backed up by a substrate or matrix. Conventionally, the fiber would lie on (or glued to) a flat substrate, and the indenter tip would penetrate the lateral surface of fiber (cylindrical surface). However, such lateral indentation is difficult to extract the mechanical properties of fiber, due to the curved lateral surface of fiber and complicate effect of substrate [13]. In addition, the lateral surface of some fibers
(e.g. hair) is not smooth where the roughness comes from numerous
asperities, which act like a compliant “shell” upon indentation and
lead to a larger contact area than that resulted from a perfectly
smooth lateral surface [16]
– such a wrongly measured contact area, plus the uncertainties of
adhesion force due to asperities, can lead to a significant error of
the measured hardness and stiffness of the fiber during a conventional lateral indentation experiment [17].
In order to circumvent these difficulties, an indentation test along the axial direction and on the cross-section of the fiber (normal indentation) is desired. In order to avoid buckling, the fiber needs to be embedded in and supported by a large matrix (e.g. epoxy); multiple fibers
can be aligned in the matrix as long as their separation is
sufficiently large such that they do not affect each other upon
indentation. Once the cross-section of the composite is polished, a
normal nanoindentation test on the cross-section of the fiber can be carried out (see the schematic in Fig. 1(a)) – such technique has been applied to measure the contact stiffness of human hair and optical fiber [7] and [18].
| Full-size image (53K) |
Schematic of (a) conical indentation on a fiber/matrix system and (b) indentation load–depth (P–δ) curve obtained from indentation test.
Although
the normal indentation approach has overcome the disadvantages of
surface curvature and roughness encountered in the lateral indentation
on fibers,
the influence of matrix emerges as the primary issue of affecting the
normal indentation measurement. The finite stiffness of matrix makes
the “matrix effect” prominent in the measured indentation force–displacement behavior, that is, the measured property is a mixture of the fiber and matrix properties even at relatively shallow penetration depths [19]. Hence, the classic indentation theory [14]
based on bulk materials (which is also the default option on many
commercial nanoindenters) can cause error if one would use that to
extract the fiber properties without careful considerations. In previous experiments of the normal indentation on fibers [7] and [18],
the important contribution of matrix compliance was not considered
although in some cases the indentation depths were on the same order of
fiber/hair thickness, which may result in error of the reported fiber properties since it is expected that at such deep penetration the matrix effect may not be negligible.
While the substrate effect on the thin film indentation problem has been studied extensively [19], [20] and [21], the study of the matrix effect of normal indentation on a fiber
bounded by a matrix is still lacking. To fulfill the potential of the
normal indentation technique, a theoretical framework needs to be
established to understand and then subtract off the matrix effect from
the measurement, and to obtain the intrinsic mechanical properties of
the micro- and nano-fiber. In addition, the developed method can be extended to measure the mechanical properties of fibers in a fiber-reinforced composite material, without separating fiber
from the matrix – such convenience in sample preparation may be
important in tissue engineering, where it is very time consuming and
challenging to extract a single fiber from its matrix.
fiber: using the matrix effectWith reference to Fig. 1(a), a fiber of radius R is embedded in an otherwise semi-infinite matrix, and we assume perfect bonding between the fiber and matrix. During a normal indentation test on the center of the cross-section of the fiber (Fig. 1(a)), the relationship between indentation load (P) and indentation depth (δ) can be continuously measured with the penetration of a hard indenter tip into the specimen (Fig. 1(b)). Note that due to the complicated stress and strain fields resulting from finite deformation, the P–δ
curve is only implicitly related with the material elastoplastic
properties as well as the material/system structure (e.g. the presence
of interface in the fiber/matrix
system) – the functional relationships need to be established such that
through a reverse analysis, the most important and intrinsic elastic
and plastic parameters of the fiber can be derived from the experimental data [15].
Although theoretically, one could keep the indentation depth much smaller than the fiber radius, so as to avoid the matrix effect and then apply the conventional closed-form formulae based on the Oliver–Pharr method [14], such approach might encounter difficulties for micro- and nano-fibers
for the following reasons: (1) very shallow nanoscale indentation
experiment often relies on a state-of-the-art commercial nanoindenter
that is not accessible to many users, it is more desirable to develop a
technique that could work for a much less expensive microindenter; (2)
the matrix/substrate effect on elastic property measurement persists at
very small depth especially when the matrix is compliant [19] and [20],
and thus even smaller indentation depth may be required; (3) if a
nanoscale indent is required to avoid the matrix effect in a very thin fiber, uncertainties such as surface roughness, adhesion, irregular indenter geometry and size effects may become dominant [21]
which makes the modeling very hard. To circumvent such nanoscale
effects, an indentation test with moderate penetration depth is needed
(so as to apply the well-developed continuum mechanics approach), which
would inevitably induce the matrix effect. The primary goal of this
paper is then to understand the matrix effect and obtain the intrinsic fiber properties with the presence of (or assistance of) the matrix effect.
Due
to the lack of analytical solution of the indentation problem involving
finite deformation and two-phase materials, numerical simulations based
on the finite element method (FEM) will be applied to establish
explicit relationships between the mechanical properties of the fiber and matrix (such as the elastic modulus and yield stress) and the indentation response (such as the shape factors of the P–δ curve). Inspired by the study of the substrate effect of nanoindentation on thin films [19], [20] and [21], the contribution of matrix is enhanced with increasing δ in the present study. That is, the curvature of the measured P–δ
curve is different at distinct indentation depths, and the matrix
effect can provide sufficient information for determining the
elastoplastic properties of fiber. In the following, we will first vary the fiber
and matrix properties in a wide range, such that during the forward
analysis with FEM simulations, the explicit relationships can be
established between the material properties and the indentation
response at different representative indentation depths (which
correspond to different matrix effects). An effective reverse analysis
is established such that, once the matrix modulus is known or measured,
the elastic modulus and yield stress of the fiber can be readily identified from the measured indentation P–δ
curve. The robustness of the technique is verified through numerical
indentation experiments and an error sensitivity analysis. In addition,
parallel experiments on human hair and glass fibers are carried out to examine the proposed indentation method, and the results (extracted fiber
properties) agree well with the reference properties. The framework
proposed in this study may provide an effective approach to measure the
elastoplastic properties of micro- and nano-fibers.
For the fiber/matrix system show in Fig. 1(a), the fiber is simplified as an elastic-perfectly plastic material. Such assumption is valid for ceramic/glass and high-strength metal fibers [22], [23], [24], [25] and [26], in addition, most polymer materials have negligible hardening effect [27] and [28]; moreover, for these materials their tensile and compressive behaviors are almost symmetric, allowing the determination of fiber’s
elastoplastic properties from indentation test (which is
compression-dominant). The matrix is homogeneous and isotropic, and it
is assumed to deform elastically – such assumption is valid for the
model epoxy matrix (which is also used in parallel experiments
described below) and for the range of fiber
properties and indentation depths studied in this paper (see below),
where our FEM simulation results verified that at the maximum
penetration, the peak Mises stress induced in epoxy is insufficient to
cause matrix yielding.
Since the matrix is much larger than the fiber, the matrix’s Young’s modulus, Em, can be easily measured by using the uniaxial tension test. Alternatively, the Oliver–Pharr method [14]
may be employed to obtain the stiffness of matrix through indentation
(where the matrix can be taken as a bulk material) which is also used
in our parallel experiment in this paper. If the matrix mechanical
property is known, there are only two unknown material parameters: the fiber’s Young’s modulus, Ef and the fiber’s yield stress, σf,
which can be mathematically determined by using two independent
equations. Meanwhile, more equations may be employed and such
redundancy may help to improve the numerical reverse analysis result
through optimization. The Poisson’s ratio, normally regarded as a minor
factor during indentation [19], is set to be 0.3 for both fiber and matrix.
To simplify the analysis, the indenter tip is assumed as a rigid cone with a half apex angle α=70.3°, which has the similar ratio of cross-section area to indentation depth as the widely used Berkovich indenter [15].
It has been reported that the conical indentation has almost the same
loading and unloading curves (with less than 1% difference in general)
as the pyramidal indentation as long as both types of indenters have
the same ratio of cross-section area to indentation depth [15]; therefore, conical indenter will be used in this study to take advantage of axisymmetry (see Section 2.3). Coulomb’s friction law is applied between indenter and fiber
surface with a friction coefficient of 0.1. The friction is a minor
factor in indentation as long as the friction coefficient is small,
which has been verified by our FEM analysis and other groups [15]. We assume perfect bonding between the fiber and matrix during the experiment.
Fig. 1(b) shows a typical P–δ curve, where Pm and δm
are the maximum indentation load and indentation depth, respectively.
From the principle of dimensional analysis, the dimensionless curvature
of the loading curve can be expressed as:which may be regarded as shape factors reflecting the matrix effect at various penetrations (e.g. δ1,δ2,δ3). To solve the
fiber properties (Ef,σf),
the required two independent equations can be provided by the distinct
matrix effects at different indentation loads/depths with (P1,δ1) and (P2,δ2), see Fig. 1(b) where δ1=δm:The dimensionless indentation depths δ1/R and δ2/R are chosen such that their corresponding matrix effects are distinct (δ1/R=0.25 and δ2/R=0.075 in this paper, see Section 3.1 for justification). F1 and F2 are dimensionless functions that can be determined from extensive FEM simulations in forward analysis, see Section 3.1. Practically, one could use more data points (such as (P3,δ3) in Fig. 1(b))
to establish additional (redundant) functional forms of a different
matrix effect, which can help to improve the accuracy of the reverse
analysis, discussed in Section 3.2.4.
Note that with the current normal indentation framework (where the fiber is elastic-perfectly plastic), we do not require the use of unloading P–δ curve from the experiment. In fact this is advantageous based on the following reasons:
(1) Since we do not have to perform unloading at the maximum penetration (δ1/R=0.25 in this study), the fiber radius does not need to be pre-determined accurately and a moderate scatter of fiber radius is allowed. One can indent sufficiently deep to make sure that the maximum depth is above 1/4 of the radii of all fibers,
without the need to withdraw the indenter at a particular moment. This
extends the applicability of the proposed technique and would be
particular welcome for automated test.
(2) In a nanoindentation test, stress relaxation is often observed at the maximum load when unloading just occurs, since both the strain and the strain rate are very large just beneath the indenter (and the loading rate is usually finite during the experiment). The sudden withdraw of indenter may cause oscillations in the measurement and holding at the maximum load may also lead to uncertainty in the measured quantities. In addition, the initial portion of unloading is prone to thermal drift. All these factors require extra attention during the collection of unloading data [29], [30] and [31] in particular if the indentation depth is very small.
(3) In many established theoretical and numerical indentation techniques, a rigid indenter is used [14], [15], [19], [20], [21] and [25] whereas in practice, a diamond indenter tip is employed. It is well known that the unloading behavior is strongly affected by the compliance of indenter tip [14]. Therefore, in order to apply the developed indentation method based on the rigid indenter, a real P–δ curve measured from experiment must be converted to that measured by a rigid indenter – for example, our previous work on thin film indentation [21] utilizes the unloading work in the indentation analysis, which requires the full information of the unloading P–δ curve with a rigid indenter, and if the diamond tip contribution is not properly removed from the experimental data, plus any of the above-mentioned error due to the nonelastic effects in the unloading P-δ curve, a large error may occur based on error sensitivity analysis [21] and an careless user may make mistakes [32]. Since the diamond tip compliance has much less influence on the loading P–δ curve (especially at moderate indentation depth adopted in this paper), it is more convenient to circumvent such difficulty by using the arguably more reliable loading P–δ curve measured from an experiment.
The FEM simulations are carried out using commercial code ABAQUS [33]. The fiber is assumed to be a perfect cylinder and the indentation load is applied along the center axis of fiber. With these assumptions, the fiber
and matrix can be simplified as a 2D axisymmetric system instead of
using three-dimensional model and the computational cost can be
significantly reduced (note that if the fiber’s cross-section significantly deviates from circular, or if the indentation load is severely off from the fiber’s center axis, then 3D model/analysis must be carried out). The fiber
is meshed with about 20,000 4-node axisymmetric elements with reduced
integration, and the matrix includes about 15,000 elements with the
same type. The mesh density is designed such that the mesh is more
refined near to the indenter tip along both the axial and the radial
directions. In order to accurately capture the contact radius, the mesh
size is refined in the contact area to make sure more than 200 fiber elements are in contact with the indenter tip when δ2/R=0.075. The matrix radius is set to be 50 times larger than the fiber radius to simulate the semi-infinite “substrate”, and the fiber length is taken as 50 times of the fiber
radius – these dimensions ensures that the matrix is semi-infinite. The
degree of freedom along axial direction of nodes on the bottom of the fiber and matrix is fixed to represent the boundary condition when the sample is mounted on a rigid substrate.
In the present numerical experiments, the ratio of Ef/σf
is varied from about 20 to 3000 to cover a wide range of
engineering/biomaterials during the forward analysis, including human
hair and optical fiber.
Although epoxy is selected as a representative matrix material in the
parallel experiment (the typical epoxy modulus is about 1–5 GPa), the
ratio of Ef/Em
is also varied in a wide range from about 2 to 150, such that the
established framework may be applied to a wide combinations of fiber/matrix systems.
To verify the methodology developed in this study, nanoindentations were performed on the cross-section of human hair and glass fiber
using a Triboscope (Hysitron Inc.) in conjunction with a Veeco
Dimension 3100 AFM system (Veeco Metrology Group). The human hair and
glass fiber
were embedded into a special epoxy (302-3M, Epoxy Technology Inc.),
which produces no heat or thermal expansion during cure and thus does
not create any stress into the fibers. The sample was then cut with a cutting surface perpendicular to the fiber
using a water-cooled, low speed diamond saw and mechanically ground and
polished using abrasive and powders down to 50 nm. Nanoindentations
were performed on the polished cross-sections of human hairs and glass fibers using different indentation forces, with the indentation along the axial direction of the fibers. For the determination of reference mechanical properties of the fibers, small peak indentation forces ( for human hair and
for glass
fiber) were first used to generate a shallow penetration depth into the fibers (δ/R=0.004 for human hair and δ/R=0.01 for glass fiber)
so that the matrix effects can be ignored. The indentation loading
curves at moderate depths were then used as an input to derive the
mechanical properties of fiber using the established framework, with the detailed results given in Section 4.
Since our method requires the matrix modulus to be known a priori,
nanoindentations were also carried out on the epoxy matrix with a
nominal peak force of
to determine its mechanical properties. For all the indentation tests,
the peak indentation force was applied in 5 s and then totally released
in the same time after 5 s holding at the peak force except a fast
unloading (2 s) for epoxy matrix to minimize the creep effect (note
that the technique proposed in this paper for determining
fiber properties does not require unloading).
Unlike in a homogeneous bulk material where the loading curvature C=P/δ2 is a constant during sharp indentation [15], the stress field and plastic flow beneath the indenter tip will be disturbed by the presence of the matrix in a fiber/matrix system, and thus the loading curvature is dependent with the indentation depth (Fig. 1(b)). For a compliant matrix, Ef/Em>1, the loading curvature continuously decreases with the increase of indentation depth; whereas for a stiff substrate, Ef/Em<1,C increases with the increasing δ. In this paper, we choose δ1/R=0.25 and δ2/R=0.075
– for the wide material space adopted in this paper, these two
indentation depths yield sufficiently distinct matrix effects. By
varying Ef/σf from 50 to 3000, and Ef/Em from 2 to 150, the normalized loading curvatures and
are measured from the numerical indentation experiments, based on which the dimensionless functions F1 and F2 (Eqs. (2) and (3)) are fitted.
Due to the wide range of material properties involved, it is more practical to change Eqs. (2) and (3) to logarithmic form:where ξ=ln(Ef/σf) and η=ln(Ef/Em). The fitting error is less than 1% for the material space considered, with details given in the Appendix A.
The functions f1 and f2 are visualized as two three-dimensional continuous surfaces in Fig. 2: their difference denotes the distinct matrix effect at δ1/R=0.25 and δ2/R=0.075. The normalized loading curvatures increase as Ef/σf decreases or Ef/Em increases. The presence of matrix has a remarkable effect on the normalized indentation load when Ef/σf is large (i.e. more plastic materials), whereas the matrix effect is smaller for materials with smaller Ef/σf (more elastic materials). At the maximum penetration, the plastic flow is still constrained in the small fiber and thus the plastic fibers are more sensitive to the presence of matrix. In addition, a larger difference between the fiber/matrix
elastic moduli will take more advantage of the matrix effect. These two
functions relate the indentation response with the fiber elastoplastic properties as well as fiber/matrix elastic mismatch; once calibrated via FEM, they can be employed in the reverse analysis for determining the fiber properties, discussed next.
| Full-size image (29K) |
The surfaces of fitting functions (f1,f2) obtained from the forward analysis.
For an elastoplastic fiber embedded in an elastic matrix with known matrix modulus, in order to extract the intrinsic material parameters of the fiber, (Ef,σf), the following procedures are carried out:
(1) A normal indentation test is performed (as sketched in Fig. 1(a)), and two different loading curvatures, and
are measured.
(2) During the reverse analysis, the postulated fiber properties (Ef,σf) are varied over a large range and substituted into Eqs. (4) and (5) along with the measured indentation data – the combination of (Ef,σf) that minimizes the total error in Eqs. (4) and (5) represents the identified elastoplastic properties of the fiber. In other words, the identified (Ef,σf) will have the least square deviation e:A
practical issue is that multiple local minima may occur during the
numerical reverse analysis procedure. However, according to our recent
work [15]
the solution of film indentation with moderately deep penetration depth
is unique and we expect such conclusion is transferable to
fiber
normal indentation. Therefore, if one indeed encounters multiple local
minima, the corresponding candidate material properties are substituted
back into Eqs. (4) and (5) to examine if the theoretical loading curvatures agree with those measured – finally, the correct solution of fiber properties should be able to reproduce the loading P–δ curve measured from experiment. Such numerical iteration does not need the performance of any additional FEM simulations.
The effectiveness of the reverse analysis procedure is first examined by numerical indentation tests covering a wide range of fiber material parameters. In the numerical experiment, the fiber properties (Ef,σf)
used in simulation (termed as the input properties, or true solution)
are different than those employed in the forward analysis (Section 3.1). The fiber
is embedded in an epoxy matrix with a representative reference Young’s
modulus of 2 GPa (which does not equal to the real material used in
parallel experiment elaborated in Section 4;
here we purposely use a different matrix modulus than the one used in
experiment to test the robustness of our proposed technique). From the
data obtained from such numerical test, the reverse analysis is carried
out following the procedures described in Section 3.2.1, and the identified elastoplastic properties of fiber are compared with the input properties in Fig. 3.
| Full-size image (11K) |
The
comparison between the input material properties (end) and identified
material properties (head) from the reverse analysis of numerical
indentation experiments: the fiber/matrix combinations are varied in a large range.
In Fig. 3, for each fiber material, the end of the arrow shows the value of input parameters, and the head of arrow represents the identified fiber properties from the reverse analysis. For the examples shown in Fig. 3, the maximum deviation of the identified Ef/Em is about 14%, and that occurs when both Ef and σf are large. The maximum deviation of the extracted Ef/σf is about 6%, which happens to a sample with both relatively small values of Ef and σf. Overall, the identified fiber elastoplastic properties match well with the true solution.
Besides
numerical verification over a wide material space, we also focus on
human hair/epoxy system as it is used in parallel experiments. The
reported elastoplastic properties of human hair vary from [7], [34] and [35]: and
. Within such range, four different human hair examples are selected with input properties
, and
, respectively. The comparisons between the identified elastoplastic properties (head) and true solutions (end) are plotted in Fig. 4 as red arrows. The maximum deviations of Ef/Em and Ef/σf
are about 8% and 3%, respectively. Thus, the proposed normal
indentation techniques can be effectively employed to measure the
elastoplastic properties of human hair.
| Full-size image (8K) |
The comparison between the input material properties (end) and identified material properties (head) from the reverse analysis of numerical indentation experiments: specified for human hair/epoxy composite. The red arrow is based on two functions (4) and (5), and black arrow is based on three functions (4), (5) and (11). (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)
In a numerical indentation test, the measured data (P1,P2,δ1,δ2) cannot coincide exactly with the fitting functions (Fig. 2, or Eqs. (4) and (5))
established during the forward analysis, which will cause certain
numerical error in the reverse analysis. In addition, small errors
might also occur in the measurement of (P1,P2,δ1,δ2)
during instrumented indentation experiments (due to either random noise
or “systematic” calibration errors), which will also affect the
accuracy of the fiber
properties extracted from the reverse analysis. The error sensitivity,
i.e. how sensitive the accuracy of the reverse analysis is to these
errors, is very important for examining the robustness of the proposed
indentation framework.
A technique initial reported by Prchlik [36] is expanded to our elastoplastic fiber/elastic matrix system. The following equations can be obtained by differentiating the fitting functions (4) and (5):
Here,
each differential term represents the error of that variable. The error
of the substrate modulus is not considered since it is assumed known in
this paper. From Eqs. (7) and (8), the variations (errors) of the fiber properties (Ef,σf) can be explicitly expressed as a function of the perturbations (errors) of indentation measurements (P1,P2,δ1,δ2):
The dimensionless coefficients αi,βi(i=1–4)
can be used to represent the error sensitivity of a material parameter
corresponding to a particular error of one indentation data. For
example, if the relative error of the measured maximum indentation load
P1,dP1/P1=1%, the corresponding error of the identified Ef from the reverse analysis will be α1%. In other words, α1 is the error sensitivity factor of Ef corresponding to the perturbation of P1. If the coefficients (αi,βi) are small, then the identified fiber
elastoplastic properties are relatively insensitive to the error of the
indentation measurements, and thus the reverse analysis is robust.
We illustrate the error sensitivities of two types of important fibers: human hair and glass fiber. The representative material properties for the human hair are and
, and the typical elastoplastic parameters of a glass
fiber are and
. Based on the results of computed error sensitivity factors given in Table 1,
since these coefficients are relatively small, the proposed technique
can be considered as quite robust against the perturbation of
experimental error for these two categories of applications.
In the above analysis, two independent functions representing different matrix effects are employed to solve the fiber
elastoplastic properties via a reverse analysis. Theoretically, the
accuracy of the method can be further improved by the redundancy, i.e.
by employing more similar functions in the forward and reverse
analyses. With reference to Fig. 1(b), one can choose one more data point at (P3,δ3), with δ3/R=0.15. And the third dimensionless function can be described as:Note that this function is not completely independent with Eqs. (4) and (5). The function f3 is also given in Appendix A. In the reverse analysis, Eq. (6) can be rewritten as:
With one more functional constraint, the least square deviation may coincide better with the real material properties.
For the four hair examples used in Section 3.2.2, the redundancy analysis (Eq. (12)) is applied to recalculate the fiber
properties from the numerical indentation experiments, and the
comparisons between the identified parameters vs. input values are
shown in Fig. 4
as black arrows. With the addition of the redundant equation, there is
no significant improvement of the deviations of the identified Ef/Es and Ef/σf. Therefore, in practical applications, the two functions (Eqs. (4) and (5)) and their reverse analysis (Eq. (6)) proposed in Sections Sections 3.2.1 and 3.2.2 may be sufficient.
fiber and human hairThe parameters and reference properties of glass fiber and human hair are shown in Table 2,
whose radius was measured from microscopy and mechanical properties
were measured from careful shallow nanoindentation tests (Section 2.4),
and used as benchmark for examining the indentation technique proposed
herein. The representative AFM images of indents on glass fiber and human hair are given in Fig. 5. Next, we measure the modulus and yield strength of these fibers by carrying out deep indentation tests on the cross-sections of the fibers embedded in an epoxy matrix, and then use the framework proposed above to deduce the fiber’s elastoplastic properties via the reverse analysis. The Young’s modulus of the epoxy matrix was , measured also from the shallow nanoindentation test (Section 2.4).
| Full-size image (44K) |
(a) AFM image of an indent on a glass fiber; (b) AFM image of an indent on a human hair.
Fig. 6(a) and (b) show the indentation load–depth (P–δ) curves on three distinct glass fibers and human hairs, respectively. By input the shape factors of loading P–δ curves at into the functions built in the forward analysis, their elastoplastic properties can be identified from the reverse analysis:
and
for glass
fibers; and and
for human hair. These results are within the ranges reported in the literature [7], [34] and [35]. The difference between the average identified properties and the reference properties is less than 8% for glass
fiber
and less than 4% for human hair, respectively, and thus the proposed
indentation technique is essentially robust. It is quite interesting
that the yield strength of human hair is comparable with some of the
strongest polymers used in industry [37], whose mechanism is worth investigating in the future. We note that although in our current study, the reference fiber properties can be measured from shallow nanoindentations because the glass fiber
and human hair employed in this study were not very thin, the proposed
indentation technique which relies on the matrix effect at moderately
deep indentation depth should have wider applications especially in
cases where shallow nanoindentations are not reliable or available (as
mentioned in Section 1.2).
| Full-size image (51K) |
Indentation load–depth (P–δ) curves for (a) glass fibers with epoxy matrix; (b) human hair with epoxy matrix; the data is used in reverse analysis to obtain glass fiber and human hair properties in Table 2.
In this paper, we propose a framework of indentation technique, which measures the elastoplastic properties of a fiber
embedded in an elastic matrix via normal indentation. The method is
applicable to a range of polymer, glass, ceramic, and metal fibers. The matrix effect is investigated during the forward analysis and the intrinsic fiber
properties are extracted via a reverse analysis. The proposed method
focuses on studying the matrix effect, utilizing such effect, and
finally subtracting off such effect to obtain the fiber’s
mechanical properties. During the analysis, only the loading curve of
an indentation test is employed, which may circumvent the common
sources of errors encountered during the unloading measurement and thus
increases the potential applicability.
From the numerical forward
analysis over a large space of material properties, the relationships
between the material properties and the indentation parameters are
established by using extensive finite element simulations. Based on
these relationships, after an indentation test is carried out on the
sample fiber
embedded in an elastic matrix, the data at two different indentation
depths can be measured that correspond to different matrix effects, and
then the fiber
properties can be determined from the reverse analysis. The
effectiveness of the reverse analysis is verified by both the numerical
experiments spanning a wide range of fiber/matrix
combinations and those of the human hair/epoxy system. In addition, an
error sensitivity analysis is carried out, which shows that the current
method is essentially insensitive to the perturbations of the measured
indentation load and depth, and thus it is quite robust for determining
the elastoplastic properties of micro- and nano-fibers. We carried out experiments on both glass fibers and human hair, and the effectiveness of the proposed indentation technique is validated.
In the future, the present work may be expanded to inhomogeneous coaxial fibers (e.g. multi-layer optical fibers
or coated hair) – again, the different indentation depths will induce
the different multi-layer effects and those effects are quite
controllable (as opposed to the conventional lateral indentation where
the outmost layer is overly dominant during the test). Another future
research direction is to incorporate the strain hardening effect of the
fiber and/or matrix, as well as possible debonding between the fiber and matrix. The framework can also be extended to the fiber-reinforced composites if the fiber spacing is sufficiently apart; otherwise, the interaction among the fibers needs to be considered, which is still possible [23] by using more complicated formulations.
The work of G.C. and X.C. was supported by NSF CMS-0407743 and CMMI-0643726. Z.H.X. and X.L. were supported by NSF EPS-0296165 and CMMI-0653651.
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With ξ≡ln(Ef/σf) and η=ln(Ef/Em), the dimensionless functions of the normalized indentation curvatures are:with
and
with
The coefficients in (14) and (16) are given in Table A1.
| Coefficients in Eq. (14) | Coefficients in Eq. (16) | Coefficients in Eq. (17) | |||
|---|---|---|---|---|---|
| p1 | 3.18880 | q1 | o1 | 4.37145 | |
| p2 | −1.01670 | q2 | o2 | −1.30194 | |
| p3 | −3.22878 | q3 | o3 | −6.48993 | |
| p4 | −0.02700 | q4 | o4 | −0.05701 | |
| p5 | −0.39386 | q5 | o5 | −0.01768 | |
| p6 | 0.30349 | q6 | o6 | 0.63007 | |
| p7 | 1.02616 | q7 | o7 | 3.25744 | |
| p8 | −0.02821 | q8 | o8 | 0.00009 | |
| p9 | −0.14069 | q9 | o9 | 0.01213 | |
| p10 | 0.08576 | q10 | o10 | −0.02115 | |
| p11 | 0.54120 | q11 | o11 | −0.27911 | |
The redundant Eq. (11) can be fitted as:
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