# Reverse Analysis for Determining the Mechanical Properties of Zeolite Ferrierite Crystal

Authors: J. Lin; X. Y. Niu; X. F. Shu
Journal: Journal of Nanomaterials
 (2008)

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## Abstract

In order to explore more mechanical properties of zeolite Ferrierite (FER) single crystal, a method of determining its mechanical properties—nanoindentation reverse analysis—was obtained based on the nanoindentation experiment and numerical simulations, and this will be presented in this paper. The yield stress and the characteristic work-hardening rate were gained if its stress-strain relation was a bilinear constitutive relation. The mechanical parameters obtained by reverse analysis have been compared with ones gained by nanoindentation finite-element numerical simulations.


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## Body

## 1. Introduction

Nanoindentation experiments and finite-element numerical simulations
are useful methods for the characterization of mechanical properties of thin
film and very small-scale materials. The hardness and elastic modulus of
micrometer-sized volumes can be known from a loading-unloading curve
in the whole indented depth range according to the calculating method of Oliver and Pharr [1]. The mechanical behavior of
single crystalline hydroxyapatite was examined through instrumented nano- and microindentation
experiments on prism and basal planes by Viswanath et al. [2]. Numerical
simulations can give much information which is difficult to obtain from the nanoindentation experiment such as constitutive
relation, yield stress, and yielded zone of materials though nanoindentation progress
is very complex. In the recent years, Bolshakov et al. [3] investigated the
influences of stress on the measurement of mechanical properties by both
nanoindentation experimental studies and nanoindentation finite element
simulations using special specimens of aluminum alloy 8009. Jayaraman et al.
[4] measured and modeled the mechanical properties of the known 1070 steel by
nanoindentation tests and finite element method, respectively. The
results showed good agreement with the properties of the material. A work that was to combine
numerical simulation and nanoindentation for determining mechanical properties
of single crystal copper at mesoscale was done by Liu et al. [5]. In
order to quantify the deformation characteristics of bulk metallic glass, Vaidyanathan et al. [6] carried out
instrumented sharp indentation experiments at nano- and microlength scales. In the
same time, detailed three-dimensional finite element simulations of
instrumented indentation formulated an overall constitutive response. In addition, large deformation finite element computations were carried
out for 76 different combinations of elastic-plastic properties for two
aluminum alloys 6061-T6511 and 7075-T651 by Dao et al. [7]. Using
dimensional analysis, forward and reverse analysis algorithms were established, elastic-plastic mechanical parameters of the
two materials were obtained, and the computational results were compared with
experimental data. Liu et al.
[8] determined the mechanical properties
of metallic foams with eighty different material parameters according to
reverse analysis methods based on nanoindentation finite element simulations. Stauss et al. [9] obtained the stress-strain
behavior of small devices such as thin films, coatings, and microelectromechanical
systems by a reverse analysis of load-displacement data received from
nanoindentation experiments. Y.-T. Cheng and C.-M. Cheng [10, 11], Capehart and Y.-T. Cheng [12] used dimensional analysis and finite element calculations to derive
several scaling relationships for conical indentation into elastic-perfectly plastic solids. They revealed the general relationships
between hardness, contact area, initial unloading slope, and mechanical
properties of solids. In 2004, Y.-T. Cheng and C.-M. Cheng provided an overview of the
basic concepts of scaling and dimensional analysis and reviewed works of applying
these concepts to modeling instrumented indentation measurements [13]. These methods are helpful
as a guide to determine the mechanical properties of materials including small
sizes.  Kusano and Hutchings of the University
of Cambridge made use of the method of Cheng to achieve hardness and modulus of
carbon nitride films and silicon substrates. The data analysis of this method
was compared with other methods. As  Kusano said, “The method described by
Y.-T. Cheng and C.-
M. Cheng appears to provide the most reliable values for hardness and
modulus for carbon nitride films” [14].Zeolites are microporous crystalline solids with
well-defined structures. It is well worth knowing their mechanical properties for
their strength of design in their comprehensive applications. Because of strong
compression and shear stress, their physical and chemical properties have been
thoroughly investigated. Wang et
al. [15, 16] measured to find Young’s
modulus of zeolite single
crystal ZSM-5 (MFI) averaging about 200 μm
using a microdeformation tester made by themselves in 2002. Lin et
al. [17, 18] tested the hardness
and elastic modulus of zeolites FER and SOD of
smaller sizes than ZSM-5 by nanoindentation experiments. Afterward, Brabec et al. [19] measured hardness and elastic modulus
of zeolite silicalite-1 crystal twins from depth-sensing indentations using
Berkovich tip in 2006. In the same year, Lethbridge et al. [20] had a typical indentation
experiment, in which they measured Young’s moduli of the zeolite single-crystal
natrolite, and comparison
with dynamic studies and simulations. In 2006, Niu et al. [21] determined bilinear elastic-plastic constitutive relation of
zeolite crystals FER and SOD.In the
present paper, based on the nanoindentation
experiment and numerical simulations, the
elastic-plastic bilinear constitutive relation of zeolite FER single crystal
was gained using a nanoindentation reverse analysis method. The values of the
mechanical parameters have been compared in between nanoindentation reverse
analysis and finite-element numerical simulations.


## 2. Nanoindentation Reverse Analysis of
Zeolite Ferrierite Single Crystal

Zeolite Ferrierite (FER) single crystal is a
medium-pore-type zeolite, containing chains of five-membered rings, which are
linked to give54 polyhedral units from which the three-dimensional
framework can be built. It contains a two-dimensional
network of 10-MR pores (4.3×5.5 Å) and 8-MR (3.4×5.5 Å) intersecting channels [22]. Zeolite FER single
crystal has been widely used as catalysts in chemistry reactions. For example, it may
be used as catalyst of n-butene isomerization [23] and also can be employed as catalyst of NOx reduction [24]. Representative image of zeolite FER single crystal
was displayed in Figure 1(a), and the framework structure was played in Figure 1(b).
From this image, we can see that the shape of zeolite FER single crystal is flaky
about 290×180μm.The image and framework structure of typical sample of zeolite FER single
crystal.
(a)
The image of zeolite FER(b)
The framework structure of zeolite FER


### 2.1. Calculation of Nanoindentation Finite-Element Numerical Simulations of Zeolite FER Single Crystal

The elastic-plastic bilinear
material model was presented. Now, we must make finite element numerical
simulation for zeolite FER single crystal before the calculation of reverse
analysis. A schematic representation of the bilinear constitutive law was shown
in Figure2 [25].Figure 2
Elastic-plastic bilinear stress-strain curve for zeolite FER single crystal.The elastic-plastic stress-strain behavior of zeolite FER single crystal was given by the expression(1)σ={Eε,ε≤ε0,σ0+ETε,ε>ε0, where E was the elastic modulus, and ET was the characteristic work-hardening rate. σ0 and ε0 were the corresponding stress and
strain values, respectively, when the material reached the yield.Because of the axisymmetrical structure of
zeolite FER single crystal, axisymmetrical material and conical rigid indenter
were used in the finite element in place of the triangular Berkovich indenter
used in the experimental study where had a real deformation field. In order to simplify the sample
model, the half-included tip angle of the conical indenter was70.3°, giving the sample depth-to-area ratio as
the Berkovich pyramid used in the experiments [25].Simultaneous control point and control line were used for
simulating the shape of indenter. The model was
determined as a square of35×35μm because
of the indentation depth 1100.2 nm when the load’s peak value in the experiment
had no influence on the result. The typical axisymmetric geometry and mesh used
in the study were
shown in Figure 3.Figure 3
The finite element mesh of the indentation process for zeolite FER single crystal.It
was known that the mechanical properties  of zeolite FER single crystal were confirmed
from the parametersE,σ0,ET in (1). Yield stress σ0 was chosen at the range of 0.067–0.938 GPa,
and there were six values. If the characteristic work-hardening rate ET was chosen at the range of 0.03–0.5 GPa,
three values were assumed as shown in Table 1. So,
eighteen group parameters were constructed, and eighteen group
load-displacement curves were obtained through calculating nanoindentation finite-element
numerical simulations in which Yuong's modulus of
zeolite FER single crystal was 10 GPa from the nanoindentation experiment [17].
Its Poisson's ratio was 0.25 for a 5% error range when the Poisson's ratio of
any material was not known for nanoindentation.Table 1
The mechanics properties of zeolite FER single crystal
used in the finite-element numerical
simulations.
E(GPa)σ0(GPa)E•/σ0σ0/E*ET(GPa)ET/E*100.067159.30.00630.030.0028100.085125.50.0080.080.0075100.11791.20.0110.50.047100.18856.760.0176100.4722.70.044100.93811.380.0879Calculating data of eighteen groupsC and hr/hm were obtained from eighteen load-displacement curves like Figure 6
through the nanoindentation finite-element numerical simulations (in Table 2). C was the
indentation curvature, a measure of the “resistance” of the material to
indentation (referring to relation 17). hr was final indentation depth after unloading,
and hm was indentation depth at peak load (referring
to Figure 6).Table 2
18 groupsC and hr/hm values through calculation of nanoindentation finite-element
numerical simulations of zeolite FER single crystal.
C(GPa)C(GPa)hr/hmhr/hmET(GPa)5.366.400.8720.8610.038.0410.430.8230.7730.0315.7919.420.6220.4330.035.456.470.8650.8450.088.0310.550.8250.7760.0815.7519.450.6260.4350.086.166.800.8150.8050.58.4210.60.7950.7550.516.1119.510.6050.4350.5


### 2.2. The Determining Connections of Yield Strengthσ0 and Characteristic Work-Hardening Rate ET for Zeolite FER Single Crystal

Formula (2) was expressed according
to the Π theorem in dimensional analysis [7] and (1) during loading: (2)Cσ0=Π1(E*σ0,ETE*). During unloading
process, it was expressed as (3)Px=E*h2Π2(hmh,σ0E*,ETE*), when Px=0,h=hr,   (3) became (4)hrhm=Π3(σ0E*,ETE*), where Px was the unloading force, and hr was final indentation depth after unloading. hm was indentation depth at peak load (referring
to Figure 6). E* was composite elastic modulus. TE*/σ0-C/σ0 curves can be obtained when the characteristic
work-hardening rate is ET=0.03 for (ET/E*)=0.0028,ET=0.08 for (ET/E*)=0.0075, and ET=0.5 for (ET/E*)=0.047 separately (see
Figure 4) according to the data of Tables 1 and 2.Dimensionless
functionC/σ0=Π1(E∗/σ0) obtained by the finite element
simulations for zeolite FER single crystal.
(a)
Dimensionless functionΠ1, when ET=0.03(b)
Dimensionless
functionΠ1, when ET=0.08(c)
Dimensionless functionΠ1, when ET=0.5When(ET/E*)=0.0028,Π1 was (5)Cσ0=Π1(E*σ0,ETE*)=23.135ln(E*σ0)−36.978. When (ET/E*)=0.0075,Π1 was (6)Cσ0=Π1(E*σ0,ETE*)=23.519ln(E*σ0)−38.054.When (ET/E*)=0.047,Π1 was (7)Cσ0=Π1(E*σ0,ETE*)=26.5ln(E*σ0)−46.878. Zeolite FERs dimensionless
function Π1 was (8)Cσ0=Π1(E*σ0,ETE*)=Aln(E*σ0)+B. Fitting the coefficient
before ln(E*/σ0,ET/E*) in functions (5), (6), and (7), we gained (9)A=75.87(ETE*)+22.94. Fitting the second item in functions (5), (6), and (7), we gained (10)B=−223.75(ETE⧫)−36.36.(E*/σ0) was in the range of 11.38–159.3 (see Table 1), and (ET/E*) was in the range of 0.0028–0.047. Therefore, (8), (9), and (10) expressed dimensionless
function Π1 for zeolite FER single crystal in total.From Tables1
and 2, we also can obtain hr/hm-σ0/E* curves (see Figure 5), when (ET/E*)=0.0028,(ET/E*)=0.0075,
and (ET/E*)=0.047, respectively.Dimensionless functionhr/hm=Π2(σ0/E*) obtained by the finite element
simulations for zeolite FER single crystal.
(a)
Dimensionless functionΠ2, when ET=0.03(b)
Dimensionless functionΠ2, when ET=0.08(c)
Dimensionless functionΠ2, when ET=0.5Figure 6
Load-displacement curve of zeolite FER in
nanoindentation experiment.When(ET/E*)=0.0028,Π2 was (11)hrhm=Π2(σ0E*,ETE*)=−0.1634ln(σ0E*)+0.0769. When (ET/E*)=0.0075,Π2 was (12)hrhm=Π2(σ0E*,ETE*)=−0.1584ln(σ0E*)+0.0948. When (ET/E*)=0.047,Π2 was (13)hrhm=Π2(σ0E*,ETE*)=−0.1426ln(σ0E*)+0.1311. Zeolite FERs dimensionless function Π2 was (14)hrhm=Π2(σ0E*,ETE*)=Aln(σ0E*)+B. Using the same method previously, fitting the coefficient before ln(E*/σ0,ET/E*) and the second items in functions (11), (12),
and (13), respectively, we gained (15)A=−15.02(ETE*)2+1.22(ETE*)−0.1667,A=−65.37(ETE*)2+4.48(ETE*)+0.065,(σ0/E*) is in the range of 0.0063–0.0879 (see Table 1), and (ET/E*) is in the range of 0.0028–0.047. As a result, (14), (15) have expressed
dimensionless function Π2 for zeolite FER single crystal in total.


### 2.3. The Calculation of Mechanical Parameters of Zeolite FER Single Crystal

Figure6 was load-displacement curve of zeolite FER that
Lin et al. [17] acquired from nanoindentation experiment. During
loading, the curve generally followed the relation [7] (16)P=Ch2, where P represented the load, and C was the indentation curvature which was a
measure of the “resistance” of the material to indentation. C and hr/hm were independent quantities that can be
directly found from a load-displacement curve without any change if materials
were defined according to the discussion of
Giannakopoulos and Suresh [26]. From
the experimental curve of zeolite FER (see Figure 6), we can know independent
quantities C=9.86GPa, that is, P=9.86h2,hr/hm=0.5975,
and elastic modulus was 10 GPa from the literature [17]. According to
contact mechanics and Berkovich indenter (three-sided pyramid) being made of
diamond material, the expression of composite elastic modulus was seen as follows: (17)E*=E1−γ2=101−0.252=10.67, where γ was the
material’s Poisson's ratio.From formula (8), we can get (18)9.86×109σ0=Aln(10.67×109σ0)+B, and from expression (14), we can get (19)0.5975=Aln(σ010.67×109)+B.Two-group
results of yield stressσ0 and the characteristic work-hardening rate ET were been obtained by solving the group
equations (9), (10), (18) and (15), (19) using MATLAB program: (20)σ0=0.1509,σ0=0.5227,ET=0.4318,ET=−7.2016.   In the two results, only σ0=0.1509,ET=0.4318 can meet the demand because 3.2<ln(E*/σ0)<5 is
according to Figure 4 or Table 1. So yield
stress σ0 and the characteristic work-hardening rate ET of zeolite FER single crystal were 0.1509 GPa
and 0.4318 GPa, respectively, using nanoindentation reverse analysis method.Therefore, the
stress-strain relation was(21)σ={10ε,ε≤0.018,0.1509+0.4318ε,ε>0.018.


### 2.4. The Comparison of Zeolite FER Single Crystal between the Reverse Algorithm and the Finite-Element Numerical Simulations

We know that the bilinear constitutive law of zeolite FER single crystal
is shown in (22) through nanoindentation finite-element numerical simulations
[21]: (22)σ={10ε,ε≤0.018,0.18+0.5ε,ε>0.018.This relation was compared with (21), and it was
known that the yield strength and the characteristic work-hardening rate were approximated
in the finite element simulation to the ones in the reverse algorithm. This
phenomenon accounted for the reliability of these two calculating methods of
determining mechanical properties of zeolite FER single crystal.Though hardness and elastic modulus can be obtained alone in the nanoindentation
experiment, the question of whether the stress-strain relationships of FER
single crystal can be uniquely determined by matching the calculated loading
and unloading curves with that measured experimentally remains to be
investigated. The possibilities of using several conical indenters of different
angles to obtain stress-strain relationships should also be investigated both
experimentally and theoretically [27, 28]. This work will be continued.


## 2.1. Calculation of Nanoindentation Finite-Element Numerical Simulations of Zeolite FER Single Crystal

The elastic-plastic bilinear
material model was presented. Now, we must make finite element numerical
simulation for zeolite FER single crystal before the calculation of reverse
analysis. A schematic representation of the bilinear constitutive law was shown
in Figure2 [25].Figure 2
Elastic-plastic bilinear stress-strain curve for zeolite FER single crystal.The elastic-plastic stress-strain behavior of zeolite FER single crystal was given by the expression(1)σ={Eε,ε≤ε0,σ0+ETε,ε>ε0, where E was the elastic modulus, and ET was the characteristic work-hardening rate. σ0 and ε0 were the corresponding stress and
strain values, respectively, when the material reached the yield.Because of the axisymmetrical structure of
zeolite FER single crystal, axisymmetrical material and conical rigid indenter
were used in the finite element in place of the triangular Berkovich indenter
used in the experimental study where had a real deformation field. In order to simplify the sample
model, the half-included tip angle of the conical indenter was70.3°, giving the sample depth-to-area ratio as
the Berkovich pyramid used in the experiments [25].Simultaneous control point and control line were used for
simulating the shape of indenter. The model was
determined as a square of35×35μm because
of the indentation depth 1100.2 nm when the load’s peak value in the experiment
had no influence on the result. The typical axisymmetric geometry and mesh used
in the study were
shown in Figure 3.Figure 3
The finite element mesh of the indentation process for zeolite FER single crystal.It
was known that the mechanical properties  of zeolite FER single crystal were confirmed
from the parametersE,σ0,ET in (1). Yield stress σ0 was chosen at the range of 0.067–0.938 GPa,
and there were six values. If the characteristic work-hardening rate ET was chosen at the range of 0.03–0.5 GPa,
three values were assumed as shown in Table 1. So,
eighteen group parameters were constructed, and eighteen group
load-displacement curves were obtained through calculating nanoindentation finite-element
numerical simulations in which Yuong's modulus of
zeolite FER single crystal was 10 GPa from the nanoindentation experiment [17].
Its Poisson's ratio was 0.25 for a 5% error range when the Poisson's ratio of
any material was not known for nanoindentation.Table 1
The mechanics properties of zeolite FER single crystal
used in the finite-element numerical
simulations.
E(GPa)σ0(GPa)E•/σ0σ0/E*ET(GPa)ET/E*100.067159.30.00630.030.0028100.085125.50.0080.080.0075100.11791.20.0110.50.047100.18856.760.0176100.4722.70.044100.93811.380.0879Calculating data of eighteen groupsC and hr/hm were obtained from eighteen load-displacement curves like Figure 6
through the nanoindentation finite-element numerical simulations (in Table 2). C was the
indentation curvature, a measure of the “resistance” of the material to
indentation (referring to relation 17). hr was final indentation depth after unloading,
and hm was indentation depth at peak load (referring
to Figure 6).Table 2
18 groupsC and hr/hm values through calculation of nanoindentation finite-element
numerical simulations of zeolite FER single crystal.
C(GPa)C(GPa)hr/hmhr/hmET(GPa)5.366.400.8720.8610.038.0410.430.8230.7730.0315.7919.420.6220.4330.035.456.470.8650.8450.088.0310.550.8250.7760.0815.7519.450.6260.4350.086.166.800.8150.8050.58.4210.60.7950.7550.516.1119.510.6050.4350.5


## 2.2. The Determining Connections of Yield Strengthσ0 and Characteristic Work-Hardening Rate ET for Zeolite FER Single Crystal

Formula (2) was expressed according
to the Π theorem in dimensional analysis [7] and (1) during loading: (2)Cσ0=Π1(E*σ0,ETE*). During unloading
process, it was expressed as (3)Px=E*h2Π2(hmh,σ0E*,ETE*), when Px=0,h=hr,   (3) became (4)hrhm=Π3(σ0E*,ETE*), where Px was the unloading force, and hr was final indentation depth after unloading. hm was indentation depth at peak load (referring
to Figure 6). E* was composite elastic modulus. TE*/σ0-C/σ0 curves can be obtained when the characteristic
work-hardening rate is ET=0.03 for (ET/E*)=0.0028,ET=0.08 for (ET/E*)=0.0075, and ET=0.5 for (ET/E*)=0.047 separately (see
Figure 4) according to the data of Tables 1 and 2.Dimensionless
functionC/σ0=Π1(E∗/σ0) obtained by the finite element
simulations for zeolite FER single crystal.
(a)
Dimensionless functionΠ1, when ET=0.03(b)
Dimensionless
functionΠ1, when ET=0.08(c)
Dimensionless functionΠ1, when ET=0.5When(ET/E*)=0.0028,Π1 was (5)Cσ0=Π1(E*σ0,ETE*)=23.135ln(E*σ0)−36.978. When (ET/E*)=0.0075,Π1 was (6)Cσ0=Π1(E*σ0,ETE*)=23.519ln(E*σ0)−38.054.When (ET/E*)=0.047,Π1 was (7)Cσ0=Π1(E*σ0,ETE*)=26.5ln(E*σ0)−46.878. Zeolite FERs dimensionless
function Π1 was (8)Cσ0=Π1(E*σ0,ETE*)=Aln(E*σ0)+B. Fitting the coefficient
before ln(E*/σ0,ET/E*) in functions (5), (6), and (7), we gained (9)A=75.87(ETE*)+22.94. Fitting the second item in functions (5), (6), and (7), we gained (10)B=−223.75(ETE⧫)−36.36.(E*/σ0) was in the range of 11.38–159.3 (see Table 1), and (ET/E*) was in the range of 0.0028–0.047. Therefore, (8), (9), and (10) expressed dimensionless
function Π1 for zeolite FER single crystal in total.From Tables1
and 2, we also can obtain hr/hm-σ0/E* curves (see Figure 5), when (ET/E*)=0.0028,(ET/E*)=0.0075,
and (ET/E*)=0.047, respectively.Dimensionless functionhr/hm=Π2(σ0/E*) obtained by the finite element
simulations for zeolite FER single crystal.
(a)
Dimensionless functionΠ2, when ET=0.03(b)
Dimensionless functionΠ2, when ET=0.08(c)
Dimensionless functionΠ2, when ET=0.5Figure 6
Load-displacement curve of zeolite FER in
nanoindentation experiment.When(ET/E*)=0.0028,Π2 was (11)hrhm=Π2(σ0E*,ETE*)=−0.1634ln(σ0E*)+0.0769. When (ET/E*)=0.0075,Π2 was (12)hrhm=Π2(σ0E*,ETE*)=−0.1584ln(σ0E*)+0.0948. When (ET/E*)=0.047,Π2 was (13)hrhm=Π2(σ0E*,ETE*)=−0.1426ln(σ0E*)+0.1311. Zeolite FERs dimensionless function Π2 was (14)hrhm=Π2(σ0E*,ETE*)=Aln(σ0E*)+B. Using the same method previously, fitting the coefficient before ln(E*/σ0,ET/E*) and the second items in functions (11), (12),
and (13), respectively, we gained (15)A=−15.02(ETE*)2+1.22(ETE*)−0.1667,A=−65.37(ETE*)2+4.48(ETE*)+0.065,(σ0/E*) is in the range of 0.0063–0.0879 (see Table 1), and (ET/E*) is in the range of 0.0028–0.047. As a result, (14), (15) have expressed
dimensionless function Π2 for zeolite FER single crystal in total.


## 2.3. The Calculation of Mechanical Parameters of Zeolite FER Single Crystal

Figure6 was load-displacement curve of zeolite FER that
Lin et al. [17] acquired from nanoindentation experiment. During
loading, the curve generally followed the relation [7] (16)P=Ch2, where P represented the load, and C was the indentation curvature which was a
measure of the “resistance” of the material to indentation. C and hr/hm were independent quantities that can be
directly found from a load-displacement curve without any change if materials
were defined according to the discussion of
Giannakopoulos and Suresh [26]. From
the experimental curve of zeolite FER (see Figure 6), we can know independent
quantities C=9.86GPa, that is, P=9.86h2,hr/hm=0.5975,
and elastic modulus was 10 GPa from the literature [17]. According to
contact mechanics and Berkovich indenter (three-sided pyramid) being made of
diamond material, the expression of composite elastic modulus was seen as follows: (17)E*=E1−γ2=101−0.252=10.67, where γ was the
material’s Poisson's ratio.From formula (8), we can get (18)9.86×109σ0=Aln(10.67×109σ0)+B, and from expression (14), we can get (19)0.5975=Aln(σ010.67×109)+B.Two-group
results of yield stressσ0 and the characteristic work-hardening rate ET were been obtained by solving the group
equations (9), (10), (18) and (15), (19) using MATLAB program: (20)σ0=0.1509,σ0=0.5227,ET=0.4318,ET=−7.2016.   In the two results, only σ0=0.1509,ET=0.4318 can meet the demand because 3.2<ln(E*/σ0)<5 is
according to Figure 4 or Table 1. So yield
stress σ0 and the characteristic work-hardening rate ET of zeolite FER single crystal were 0.1509 GPa
and 0.4318 GPa, respectively, using nanoindentation reverse analysis method.Therefore, the
stress-strain relation was(21)σ={10ε,ε≤0.018,0.1509+0.4318ε,ε>0.018.


## 2.4. The Comparison of Zeolite FER Single Crystal between the Reverse Algorithm and the Finite-Element Numerical Simulations

We know that the bilinear constitutive law of zeolite FER single crystal
is shown in (22) through nanoindentation finite-element numerical simulations
[21]: (22)σ={10ε,ε≤0.018,0.18+0.5ε,ε>0.018.This relation was compared with (21), and it was
known that the yield strength and the characteristic work-hardening rate were approximated
in the finite element simulation to the ones in the reverse algorithm. This
phenomenon accounted for the reliability of these two calculating methods of
determining mechanical properties of zeolite FER single crystal.Though hardness and elastic modulus can be obtained alone in the nanoindentation
experiment, the question of whether the stress-strain relationships of FER
single crystal can be uniquely determined by matching the calculated loading
and unloading curves with that measured experimentally remains to be
investigated. The possibilities of using several conical indenters of different
angles to obtain stress-strain relationships should also be investigated both
experimentally and theoretically [27, 28]. This work will be continued.


## 3. Conclusions

In order to determine the
mechanical properties of small size zeolites FER single crystal, a new method,
reverse analysis, was put forward. This method has been carried out with
eighteen different material parameters
for zeolite FER single crystal based on bilinear modeling. The calculative
results in conjunction with the dimensionless analysis method were used to
establish the relationship between the cone indentation behavior and the
elastic-plastic material parameters of zeolite FER single crystal. The result
was to obtain its yield stress and the characteristic work-hardening rate being 0.1509 GPa and
0.4318 GPa, respectively. Two methods between reverse analysis and finite
element simulations were compared, and the yield stress and the characteristic
work-hardening rate were consistent for zeolite FER single crystal. Obtained results
showed that the mechanical properties of zeolite FER single crystal can be unambiguously
determined assuming bilinear constitutive relations by the developed
nanoindentation reverse analysis method. Therefore, the reverse analysis method
gave a good guideline for the determination of constitutive behavior of zeolite
FER single crystal. Furthermore, this technique is a potential method for researching
mechanical properties of more zeolites and other small volume materials.



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*Source: 395738-2009-03-05.xml*