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.
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 m 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 give polyhedral units from which the three-dimensional framework can be built. It contains a two-dimensional network of 10-MR pores ( Å) and 8-MR ( 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 -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
(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 Figure 2 [25].
The elastic-plastic stress-strain behavior of zeolite FER single crystal was given by the expression where was the elastic modulus, and was the characteristic work-hardening rate. and 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 was 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 of m 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.
It was known that the mechanical properties of zeolite FER single crystal were confirmed from the parameters in (1). Yield stress was chosen at the range of 0.067–0.938 GPa, and there were six values. If the characteristic work-hardening rate 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.
Calculating data of eighteen groups and were obtained from eighteen load-displacement curves like Figure 6 through the nanoindentation finite-element numerical simulations (in Table 2). was the indentation curvature, a measure of the “resistance” of the material to indentation (referring to relation 17). was final indentation depth after unloading, and was indentation depth at peak load (referring to Figure 6).
2.2. The Determining Connections of Yield Strength and Characteristic Work-Hardening Rate for Zeolite FER Single Crystal
Formula (2) was expressed according to the theorem in dimensional analysis [7] and (1) during loading: During unloading process, it was expressed as when (3) became where was the unloading force, and was final indentation depth after unloading. was indentation depth at peak load (referring to Figure 6). was composite elastic modulus. - curves can be obtained when the characteristic work-hardening rate is for for and for separately (see Figure 4) according to the data of Tables 1 and 2.
| (a) Dimensionless function when |
| (b) Dimensionless function when |
| (c) Dimensionless function when |
When was When was When was Zeolite FERs dimensionless function was Fitting the coefficient before in functions (5), (6), and (7), we gained Fitting the second item in functions (5), (6), and (7), we gained was in the range of 11.38–159.3 (see Table 1), and was in the range of 0.0028–0.047. Therefore, (8), (9), and (10) expressed dimensionless function for zeolite FER single crystal in total.
From Tables 1 and 2, we also can obtain - curves (see Figure 5), when and respectively.
| (a) Dimensionless function when |
| (b) Dimensionless function when |
| (c) Dimensionless function when |
When was When was When was Zeolite FERs dimensionless function was Using the same method previously, fitting the coefficient before and the second items in functions (11), (12), and (13), respectively, we gained is in the range of 0.0063–0.0879 (see Table 1), and is in the range of 0.0028–0.047. As a result, (14), (15) have expressed dimensionless function for zeolite FER single crystal in total.
2.3. The Calculation of Mechanical Parameters of Zeolite FER Single Crystal
Figure 6 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] where represented the load, and was the indentation curvature which was a measure of the “resistance” of the material to indentation. and 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 that is, 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: where was the material’s Poisson's ratio.
From formula (8), we can get and from expression (14), we can get
Two-group results of yield stress and the characteristic work-hardening rate were been obtained by solving the group equations (9), (10), (18) and (15), (19) using MATLAB program: In the two results, only can meet the demand because is according to Figure 4 or Table 1. So yield stress and the characteristic work-hardening rate 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
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]:
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.