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An experimental and numerical investigation on damping capacity of nanocomposite
Materials Science and Engineering A 507 (2009) 149–154
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
An experimental and numerical investigation on damping capacity of nanocomposite M. Yadollahpour a , J. Kadkhodapour a , S. Ziaei-Rad a,∗ , F. Karimzadeh b a b
Department of Mechanical Engineering, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran Department of Materials Engineering, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran
a r t i c l e
i n f o
Article history: Received 13 September 2008 Received in revised form 25 November 2008 Accepted 7 January 2009 Keywords: Damping capacity Nanocomposite Grain size Reinforcement Finite element Modal testing
a b s t r a c t In the present study, the damping capacity of a metal–matrix nanocomposite is predicted using a micromechanical modeling approach. The model is based on finite element analysis of a unit cell, which mimics a pure metallic lattice with stiff reinforcing nanoparticles. The dissipated energy of nanocomposite is predicted numerically by applying a harmonic load on the unit cell model. The model was then validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposite. Good agreement was observed between the predicted and experimental values. Finally, a parametric study was carried out to calculate the effects of different parameters such as reinforcement volume fraction and grain size on the damping characteristics of nanocomposite. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Structural materials that exhibit high damping capacities are desirable for mechanical vibration suppression and acoustic noise attenuation. A high damping capacity is observed in materials exhibiting poor mechanical properties, for instance low yield stress or hardness. For example, it is well known that the high damping in lead is associated with a very weak tensile strength. On the contrary, steels present good mechanical properties and a low level of damping. It would therefore be of interest to search for new materials simultaneously exhibiting good mechanical properties and high damping. This is possible only when the microscopic mechanisms responsible for dissipation of the vibration energy (internal friction) are independent of the hardening mechanisms. Vibration damping is the conversion of mechanical vibration energy to thermal energy, and subsequent dissipation of the thermal energy through the volume of the material or structure. Hence, damping decreases resonance structural instability and excessive noise set up by vibrating bodies. Naturally, all materials show some inherent damping capabilities. The presence of an internal resistive force in the material ensures vibration amplitude decays with time. This resistive force is a function of atomic structure, nanostructure, stress, temperature and frequency. Damping capacities therefore
∗ Corresponding author. Tel.: +98 311 3915244; fax: +98 311 3912628. E-mail address: [email protected] (S. Ziaei-Rad). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.01.016
differ significantly from one material to the other because of the dependence of the internal resistive force on atomic characteristics and nanostructure. Vibration damping can be categorized as either material damping or structural damping. Material damping refers to vibration dissipation that occurs within the volume of a material through atomic level interactions such as elastic stretching of atomic bonds, vacancy diffusion, dislocation motion and grain boundary motion. Structural damping refers to the energy dissipation resulting from the friction at joints, interfaces and fasteners. Materials with high damping capabilities are desired from the viewpoint of vibration suppression in structures. Their inherent passive damping capabilities help in stabilizing the structure when it vibrates. Structural metals tend to have very low damping capacities. Hence, to protect metallic structures from unwanted vibrations, passive or active damping techniques are widely used. Passive damping involves the use of add-on materials with high damping capacities for vibration absorption and dissipation. Active damping relies on external devices such as actuators and sensors for vibration detection and control [1–3]. To explain the passive damping capability of material, it has to be mentioned that all real materials exhibit a departure from ideal elastic behavior, even at very small strain values. These departures, under cyclic deformation, result in irreversible energy dissipation in the materials. The causes of such loss are many, including the irreversible transfer of mechanical energy into heat, growth of cracks and other defects, and micro-plastic deformation of crystals.
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Nanocomposite materials, a new kind material, have a lot of unrevealed properties to be developed. They may exhibit increased strength/hardness, improved toughness, reduced elastic modulus and ductility and enhanced diffusivity in comparison with conventional materials. These attributes have generated considerable interest in the use of Nanocomposite for a wide variety of structural applications [4]. The energy dissipation behavior of the nanocomposites is still a fresh ground and need to be more explored. In nanocomposites, local plastic deformation may arise even if the overall external load is below the yield stress of the nanocomposites because of the grain structure. Actually, plastic deformation in nanocomposites is a main source of energy dissipation during external loading. With the development of computational materials science, materials scientists pay increasingly more attention to materials modeling. The relevance of materials modeling is fundamentally predicted on the belief that such analysis and prediction will bring about a better understanding and control of materials microstructures, which in turn are essential in the application area of processing, performance evaluation, and ultimately the design of new materials. The experimental method does not always take effect due to the complexity of the microstructure, which makes it very difficult to find the controlling factor that will play the most important role among all factors in the microstructure. In such cases, modeling can be a useful tool in effectively reducing the number of experimental trials involved in materials design and in understanding the complex phenomena causing changes in the microstructure, although it cannot replace experiments. Cai et al. [5] measured internal friction in nanocrystalline aluminum prepared by plasma evaporation and compaction as a function of temperature with five frequencies ranging from 0.3 to 3.1 Hz. They observed two relaxation peaks around 180 ◦ C. Numerical modeling was used by Wang et al. to investigate the damping behaviors of Al/SiC particulate-reinforced metal–matrix composites (PMMCs) at room temperature. Through the cell method and finite-element method, the influences of particulate shape and distribution on the damping capacity of Al/SiC(p) composite are studied [6]. A new model for calculating the damping capacity of particulate-reinforced metal–matrix composites (PMMCs) is proposed by Gu et al. [7]. Their model based on the assumption that the energy loss mainly results from the anelasticity of the particulate and matrix and the micro-plasticity of the matrix under small strain amplitude. Finite element method with a multi-particle model has been adopted. The results showed that the energy loss in the loading direction can represent the total energies consumed in the composites. Trojanová et al. [8] measured strain amplitude dependence of the logarithmic decrement on microcrystalline and nanocrystalline magnesium unreinforced and reinforced with nanoparticles of Al2 O3 , ZrO2 , and graphite. Measurements were carried out before and after step-by-step isochronal thermal treatment at increasing temperature. Their results showed that a decrease in the grain size and secondary phases increase the strain independent component of the decrement. Microstructure changes due to thermal treatment are responsible for an increase of the strain dependent component of the decrement. In a separate study, Liu et al. [9] were measured the low temperature internal friction of a variety of hydrogenated amorphous, nanocrystalline, polycrystalline, and epitaxial silicon thin films. They observed that with increasing volume of nanocrystallinity, the internal friction increases in contrast to our common expectation. The simulation results are compared with the experimental results, which show that they are consistent with each other in quality. Patel et al. [10] used a finite element based unit cell model with periodic boundary conditions to calculate the effective tan ı of a composite as a function of frequency. Their results demonstrated that the tan ı of the nanocomposite manifests important information about the extent and properties of the interphase region that
surrounds each nanoparticle and concluded that measurements of this parameter could provide a simple yet useful experimental tool to understand more about the interphase. However, the measurement of this parameter may prone to error when severe agglomeration or wide size distribution of the nanoparticles are present. Literature review shows that the finite element method is a useful numerical tool capable of predicting the damping properties of metal matrix composites. Research findings of Xu and Schamauder [11], Wang et al. [6,12–13] and Gu et al. [7,14] have been successful in studying the plastic energy dissipation in metal–matrix composites at high strain amplitudes. Although, there have been some progresses in the development of method to estimate the nonlinear elasto-plastic behavior of nanocomposites [15–20], the energy dissipation behaviors of the nanocomposites have not been involved in any of these investigations. In this paper, the finite element method is employed to predict the energy dissipation of nanocomposite material under cyclic loading. A model was developed and then validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposites. Finally, the influence of the grain size and reinforcement volume fraction on the energy dissipation was also calculated by the developed finite element model. 2. Theoretical background 2.1. The model for nanocomposite The nanocomposite contains nanocrystalline (NC) Al–Zn as grain and Al2 O3 nanoparticles as reinforcement. Considering reinforcement as an elastic material, grain and boundary phase are modeled by following assumptions. NC materials are structurally characterized by a large volume fraction of grain boundaries, which may significantly alter their physical, mechanical, and chemical properties in comparison with conventional coarse-grained polycrystalline materials. As the grain size decreases, an increasing fraction of atoms can be ascribed to the grain boundaries. Two types of atoms in the NC structures can be recognized: crystal atoms and boundary atoms. Based on the above model for NC structures, a thickness is assigned for the grain boundary zone in the micrograph which is shown in Fig. 1a. In this figure, the white grains are the reinforcement and the colored grains are representative of the matrix. The difference in orientation of matrix grains is shown in the picture by different degree of grayness. To construct the composite model which is shown in Fig. 1b, some idealization on the shape and arrangement of grains was carried out. The unit cell model considers the harmonic repeating part of multi-cell model. To construct the unit cell model which is shown Fig. 1c, some simplification on the crystallographic orientations of real microstructure and boundary phase arrangement was carried out. For the crystal phase, the elasto-plastic behavior taking grain size into account by Hall–Pitch relation was used, i.e. p
(g)
p
e = y + hg εe (g)
y
(1)
= y∞ + kd−1/2
(2)
p
p
In Eq. (1), e is von Mises’ effective plastic stress and εe is effective plastic strain. The constant hg is the coefficient of hardening. (g)
Eq. (2) relates yield stress (y ) with the grain size by Hall-Pitch relation. In this equation k is a material constant, and d is a grain size. As a general estimation, the boundary thickness is considered independent of grain size with a constant size. As the grain size
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151
Fig. 1. Derivation of composite model from micrograph.
decreases, the number of grains which take place in a specific area of model increases. Thus, boundary becomes proportionally larger. Based on molecular dynamic (MD) simulation of NC material, Schwaiger et al. [21] suggested a value of 2.5–3.5 nm for boundary phase thickness. The value of 2.5 nm is taken in this study. For the boundary, a quasi-amorphous behavior was assumed. The quasi-amorphous behavior of a metal was modeled using an elastic perfectly plastic curve. In this study, we assume the strength of the boundary phase to increase up to an upper limit and then to stay at that level. The upper limit is set to be the strength of the material in the amorphous state, i.e. am . The stress–strain relation for amorphous materials is considered as ideal plastic behavior, and the assumption of ideal plasticity for the amorphous state is stated to be physically reasonable [17]. We, therefore, model the upper limit in the plastic response of the boundary phase as perfect plasticity. It is generally accepted that the yield stress of amorphous alloys ranges from Eam /50 to Eam /80, while the elastic modulus of the amorphous state, Eam , is approximately 60–75% of that of the corresponding equilibrium crystalline alloy [18].
Fig. 2. The technique used for calculation of W and W.
2.2. Mathematical model for damping capacity
3. Numerical predictions
The energy dissipation ability of the composite material is evaluated by the ratio of the dissipated work by the material W to the maximum of elastic energy stored in the cyclic loading process, and we use to represent the ratio:
Simulation is carried out for nanocrystalline Al–Zn and Al2 O3 as reinforcement. Nanocrystal grain size is approximated to be 40 nm and the nanocomposite contains 14% volume fraction of reinforcement. Fig. 3, depicts the boundary conditions which are applied for modeling of the tensile harmonic loading. The graph clearly depicts the grain, the boundary phase and reinforcement. The boundary conditions required to simulate the damping capacity are assumed as follows:
=
W W
(3)
where W can be written as
W =
dε
(4)
which is the area surrounded by the cyclic stress–strain curve. W is usually expressed as follows:
• For y = 0 and x = 0 surface, the symmetric boundary condition is used. The boundary condition limits displacement in normal direction.
ε0
W=
dε
(5)
0
where ε0 is the maximum strain during the loading process (Fig. 2) while and ε are the macroscopic average stress and strain of the model (Fig. 1). The loading process is quasi-static because what we study here is elasto-plastic deformation. In scientific investigations, the inverse quality factor Q−1 , is widely used to measure the damping capacity of material, which has the relationship with as Q −1 =
2
(6)
With the conception of the energy dissipation and the equations, it is possible to simulate the damping capacity Q−1 , of a material by calculating the area of the hysteresis loop [6–7,11–14].
Fig. 3. The finite element model of the nanocomposite.
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Table 1 Material properties used for simulation [14,20].
E (GPa)
y∞
(MPa) √ k (GPa nm) hg (GPa)
Grain
Boundary phase
Al2 O3 reinforcement
68.3 0.3 58 1.265 2.08
50 0.3 650 – –
380 0.22 – – –
• For the right surface, all nodes on the surface have the same displacement in the direction normal to the surface (i.e., the surface remains un-deformed). This constraint is important in calculations of the unit cell model to avoid necking at the specimen; if this happens, the unit cell model loses its validity. • For the top surface, the surface is loaded by incrementally displacing all nodes lying on the plane in the y-direction. The load is applied along axis y on the top plane of the model and characterized by strain amplitude. The loading process is quasi-static because we focus on elastic-plastic deformation. The set of parameters for simulation is tabulated in Table 1. The damping capacity versus strain amplitude is shown in Fig. 4. The damping curve is made up of a strain amplitude-independent part and a strain amplitude-dependent part. The former reflects the contribution from the an-elasticity of constituents, while the latter mainly corresponds to the plasticity. The critical strain amplitude means the onset of micro-plastic deformation, which depends strongly on the stress state within the material. Stress concentration at the material makes the micro-plastic deformation of grain much easier. These results can be interpreted as follows. In nanocomposites, there are three steps during the deformation process. In the first step, the grain and the boundary phase are elastic and stress concentration due to property mismatch causes damping. In the second step, the grains go to the plastic zone but boundaries remain elastic. Here plastic deformation in grains and stress concentration in grain boundaries cause damping. In the third step, both grain and boundary phase go to the plastic zone and their plastic deformation causes damping. 4. Experimental procedure 4.1. Materials and methods for production of Al–Zn/˛-Al2 O3 nanocompsite A mixture of commercial aluminum (99.7% purity and particle size of 50–70 m) and 15.8 wt.% ZnO powders (99.9% purity and particle size of 250 nm) was milled in planetary ball mill in order to produce Al–14 wt.%Zn/5 vol%Al2 O3 nanocomposite. The Mechanical Alloying (MA) was executed with a rotation speed of 600 rpm for predetermined hours without interruption, and the ball-powder mass ratio was 15:1. A mixture of aluminum and 14 wt.% zinc pow-
Fig. 4. Damping capacity curve versus strain amplitude in nanocomposite with grain size 40 nm.
Fig. 5. Transmission electron microscopy (TEM) of Al–Zn/␣-Al2 O3 nanocomposite.
ders (99.7% purity and particle size of 10–20 m) was also milled with a planetary ball mill in same condition. The produced powders was filled in a uniaxial die made of X40CrMoV51 (AISI H13) hot work tool steel. Then, the specimen was heated to 400 and 500 ◦ C and pressed at constant pressures of 400 MPa. The duration of hot pressing was 20 min. In order to avoid pores formation, the pressure on each specimen was not released until the specimen was cooled down to 300 ◦ C [22–23]. Transmission electron microscopy (TEM) picture of the nanocomposite is shown in Fig. 5. For our case study, the powder was poured into a cylindrical die and pressed regarding to the above mentioned procedure to create a cylinder with diameter of 50 mm and height of 15 mm. The object was then used in order to find out its modal and damping properties which are described in the next section. Detailed description of powder processing and bulk material preparation can be found in references [22] and [23], respectively. 4.2. Modal testing and analysis The damping factor of the specimen was determined by use of modal testing [24–26]. The measurement principle consists of recording the free vibrations of the specimen excited by tapping it with an appropriate hammer, as shown in Fig. 6. The amplitude decay as a function of time and the vibration modes were detected by an acquisition data system from B&K Company and recorded using ICATS software [27] and also by analytical relations. The test specimen was located on soft foam and tested in a free-free condition. The experimental data were obtained via hammer testing using a B&K Pulse analyzer connected to a PC. One B&K piezoelectric accelerometer (Endevco. 2222c) was attached to the structure by using wax to measure the amplitude decay of the response (Fig. 6). The test parameters were: analyses range of 0–20,000 Hz; acquisition time of 320 ms; rectangular observation window and frequency resolution of 3.125 Hz. Following the testing procedure, two types of curves were obtained: damping free vibration and frequency response function profiles. A total of three FRFs were measured and saved in appropriate format in the computers.
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153
Fig. 6. The experimental set-up for modal testing of the specimen.
The vibration test gives the free vibration damping decay and the frequency response function (FRF), simultaneously as a result. Considering a linear system of a single degree of freedom, the FRF response is the decomposition of the natural frequencies of a structure or specimen, which corresponds to a typical fingerprint identity of the vibration modes. The number of vibration peak frequencies (vibration modes) and the shape of the FRF response are a direct result of the material damping. In the second stage, the experimental FRF data were analyzed using a global multi-FRF analysis method and the results were also checked using available analytical relations [28]. The global multiFRF technique is based on a complex singular value decomposition of a system matrix expressed in terms of measured FRFs, and then on a complex eigensolution which extracts the required modal properties [27–28]. The data were analyzed by applying three runs on each FRF. Typical measured FRF is plotted in Fig. 7. An examination of the response curves shows that the shape of these curves is controlled by the amount of damping present in the system, in particular the bandwidth, that is the difference between two frequencies corresponding to the same response amplitude is related to the damping of the system. In evaluation of damping, it √ is convenient to measure the bandwidth at 1/ 2 of peak amplitude (Fig. 8). The frequencies corresponding in this bandwidth ω1 and ω2 are referred to as half-power points. It can be showed that the damping can be estimated easily by the following relation [24]: Q −1 =
ω2 − ω1 ωr
(7)
where Q−1 is the damping factor and ωr is the peak frequency in the response curves (natural frequency). In order to check the validity of the global multi-FRF method, the damping factor was also
Fig. 7. The measured frequency response function from excitation of the specimen.
Fig. 8. The half-power point technique for modal analysis.
calculated from the half-power rule (i.e. Eq. (7)). It can be seen from Table 2 that the error depends on the analysis method used for extracting the parameters from the measured data. The modal analysis will offer more accurate results as more complicated technique was used for parameter estimation. However, the difference between measured and calculated damping factor is at an acceptable level. 5. Parametric study It was mentioned previously that modeling can be a useful tool in predicting material behavior and effectively reducing the number of experimental trials involved in materials design. Having constructed and validated the finite element model, it was decided to carry out a parametric study in order to find out the effects of different parameters on the damping capacity of the nanocomposite. Damping capacity versus strain amplitude for different volume fraction of reinforcement is shown in Fig. 9. It can be observed that
Fig. 9. Damping capacity versus strain amplitude for different size of reinforcement: (a) 20 nm and (b) 30 nm.
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Table 2 Measure and estimated results for the specimen at low strain rate. Estimated parameter
Damping factor Q−1 Natural frequency (Hz) *
Experiment
Finite element method
Modal analysis
Eq. (7)
0.1177 (Error* 5.8%) 13750
0.1095 (Error = 12.4%) 13700
0.125 (Fig. 4) N/A
FEM Error (%) = Measured × 100. FEM
Finally, the influence of the grain size and reinforcement volume fraction on the energy dissipation was calculated by finite element model. It was found that by increasing the reinforcement volume fraction damping capacity versus strain amplitude would increase the damping capacity of nanocomposite. The results also indicate that decreasing the grain size enhance the damping in the nanocomposites. References
Fig. 10. Damping capacity versus strain amplitude for different grain size: (a) 30 nm; (b) 40 nm; (c) 50 nm; and (d) 100 nm.
as the reinforcement volume fraction increases, the curve shifts upward especially in larger strain amplitude. The reason is mainly due to more stress concentration in the grain and boundary phase. Damping capacity versus strain amplitude for different grain size of nanocrystalline matrix is shown in Fig. 10. It also can be observed that smaller grain size causes a downward shifting in the curve. The reason is mainly due to hardening of material and therefore reducing its damping by grain refinement. 6. Conclusions In the present study, the finite element method is employed to estimate the energy dissipation of nanocomposite material under cyclic loading. Materials and methods for production of Al–Zn/␣-Al2 O3 nanocompsite were described. Several nanocomposite cylinders were constructed and used for verifying the proposed model. The model was validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposites. The good agreement between the predicted and measured damping capacity indicate the capability of the proposed model in estimating the damping capacity of reinforced nanocomposite.
[1] T.A. Asare, Fabrication and damping behavior of particulate BaTiO3 ceramic reinforced copper matrix composites, Blacksburg, Virginia, 2004, November 19. [2] R. Schaller, J. Alloys Compd. 355 (2003) 131–135. [3] S. Sastry, M. Krishna, J. Uchil, J. Alloys Compd. 314 (1–2) (2001) 268–274. [4] D.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019–4025. [5] B. Cai, Q.P. Kong, P. Cui, H.T. Cong, X.K. Sun, Scripta Mater. 44 (2001) 1043–1048. [6] J. Wang, G. Yang, Metall. Mater. Trans. A 34A (2003) 1387–1392. [7] J. Gu, X. Zhang, M. Gu, Mater. Lett. 58 (2004) 1952–1955. [8] Z. Trojanová, P. Lukác, H. Ferkel, W. Riehemann, Mater. Sci. Eng. A 370 (2004) 154–157. [9] X. Liu, C.L. Spiel, R.D. Merithew, R.O. Pohl, B.P. Nelson, Q. Wang, R.S. Crandall, Mater. Sci. Eng. A 442 (2006) 307–313. [10] R.K. Patel, B. Bhattacharya, B. Sumit, Mech. Res. Commun. 35 (2008) 115–125. [11] D. Xu, S. Schmauder, Comput. Mater. Sci. 15 (1999) 96–100. [12] J. Wang, G. Yang, Comput. Mater. Sci. 21 (1) (2001) 9–16. [13] J. Wang, G. Yang, Mater. Sci. Eng. A303 (2001) 77–81. [14] J. Gu, X. Zhang, M. Gu, J. Alloys Compd. 381 (2004) 182–187. [15] V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleite, Acta Mater. (2002) 61–73. [16] Y.J. Wei, L. Anand, J. Mech. Phys. Sol. 52 (2004) 2587–2616. [17] M.A. Meyers, E. Ashworth, Defects Mech. Prop. 46 (1982) 737–759. [18] H.S. Kim, Y. Estrin, M.B. Bush, Acta Mater. 48 (2000) 493–504. [19] H.S. Kim, S.I. Hong, S.J. Sun Jae Kim, J. Mater. Process. Technol. 112 (2001) 109–113. [20] J.E. Carsley, J. Ning, W.W. Milligan, S.A. Hackney, E.C. Aifantis, Nanostruct. Mater. 5 (1995) 441–448. [21] R. Schwaiger, B. Moser, M. Dao, N. Chollacoop, S. Suresh, Acta Mater. (2003) 5159–5172. [22] M. Tavoosi, F. Karimzadeh, M.H. Enayati, Mater. Lett. 62 (2008) 282–285. [23] M. Tavoosi, F. Karimzadeh, M.H. Enayati, A. Heidarpour, J. Alloys Compd., in press, (2008), doi:10.1016/j.jallcom.2008.07.049. [24] D.J. Ewins, Modal Testing: Theory and Practice, Research Studies Press, 2001. [25] Handbook on Modal Testing. Dynamic Testing Agency (DTA), 1993. [26] Primer on Best Practice in Dynamic Testing, Dynamic Testing Agency (DTA), 2000. [27] ICATS Manual, Imperial College Analysis and Testing Software, June 2003. [28] M. Imregun, D.J. Ewins, 13th IMAC, Nashville, Tennessee, (1995), pp. 496–506.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
An experimental and numerical investigation on damping capacity of nanocomposite M. Yadollahpour a , J. Kadkhodapour a , S. Ziaei-Rad a,∗ , F. Karimzadeh b a b
Department of Mechanical Engineering, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran Department of Materials Engineering, Isfahan University of Technology (IUT), Isfahan 84156-83111, Iran
a r t i c l e
i n f o
Article history: Received 13 September 2008 Received in revised form 25 November 2008 Accepted 7 January 2009 Keywords: Damping capacity Nanocomposite Grain size Reinforcement Finite element Modal testing
a b s t r a c t In the present study, the damping capacity of a metal–matrix nanocomposite is predicted using a micromechanical modeling approach. The model is based on finite element analysis of a unit cell, which mimics a pure metallic lattice with stiff reinforcing nanoparticles. The dissipated energy of nanocomposite is predicted numerically by applying a harmonic load on the unit cell model. The model was then validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposite. Good agreement was observed between the predicted and experimental values. Finally, a parametric study was carried out to calculate the effects of different parameters such as reinforcement volume fraction and grain size on the damping characteristics of nanocomposite. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Structural materials that exhibit high damping capacities are desirable for mechanical vibration suppression and acoustic noise attenuation. A high damping capacity is observed in materials exhibiting poor mechanical properties, for instance low yield stress or hardness. For example, it is well known that the high damping in lead is associated with a very weak tensile strength. On the contrary, steels present good mechanical properties and a low level of damping. It would therefore be of interest to search for new materials simultaneously exhibiting good mechanical properties and high damping. This is possible only when the microscopic mechanisms responsible for dissipation of the vibration energy (internal friction) are independent of the hardening mechanisms. Vibration damping is the conversion of mechanical vibration energy to thermal energy, and subsequent dissipation of the thermal energy through the volume of the material or structure. Hence, damping decreases resonance structural instability and excessive noise set up by vibrating bodies. Naturally, all materials show some inherent damping capabilities. The presence of an internal resistive force in the material ensures vibration amplitude decays with time. This resistive force is a function of atomic structure, nanostructure, stress, temperature and frequency. Damping capacities therefore
∗ Corresponding author. Tel.: +98 311 3915244; fax: +98 311 3912628. E-mail address: [email protected] (S. Ziaei-Rad). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.01.016
differ significantly from one material to the other because of the dependence of the internal resistive force on atomic characteristics and nanostructure. Vibration damping can be categorized as either material damping or structural damping. Material damping refers to vibration dissipation that occurs within the volume of a material through atomic level interactions such as elastic stretching of atomic bonds, vacancy diffusion, dislocation motion and grain boundary motion. Structural damping refers to the energy dissipation resulting from the friction at joints, interfaces and fasteners. Materials with high damping capabilities are desired from the viewpoint of vibration suppression in structures. Their inherent passive damping capabilities help in stabilizing the structure when it vibrates. Structural metals tend to have very low damping capacities. Hence, to protect metallic structures from unwanted vibrations, passive or active damping techniques are widely used. Passive damping involves the use of add-on materials with high damping capacities for vibration absorption and dissipation. Active damping relies on external devices such as actuators and sensors for vibration detection and control [1–3]. To explain the passive damping capability of material, it has to be mentioned that all real materials exhibit a departure from ideal elastic behavior, even at very small strain values. These departures, under cyclic deformation, result in irreversible energy dissipation in the materials. The causes of such loss are many, including the irreversible transfer of mechanical energy into heat, growth of cracks and other defects, and micro-plastic deformation of crystals.
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Nanocomposite materials, a new kind material, have a lot of unrevealed properties to be developed. They may exhibit increased strength/hardness, improved toughness, reduced elastic modulus and ductility and enhanced diffusivity in comparison with conventional materials. These attributes have generated considerable interest in the use of Nanocomposite for a wide variety of structural applications [4]. The energy dissipation behavior of the nanocomposites is still a fresh ground and need to be more explored. In nanocomposites, local plastic deformation may arise even if the overall external load is below the yield stress of the nanocomposites because of the grain structure. Actually, plastic deformation in nanocomposites is a main source of energy dissipation during external loading. With the development of computational materials science, materials scientists pay increasingly more attention to materials modeling. The relevance of materials modeling is fundamentally predicted on the belief that such analysis and prediction will bring about a better understanding and control of materials microstructures, which in turn are essential in the application area of processing, performance evaluation, and ultimately the design of new materials. The experimental method does not always take effect due to the complexity of the microstructure, which makes it very difficult to find the controlling factor that will play the most important role among all factors in the microstructure. In such cases, modeling can be a useful tool in effectively reducing the number of experimental trials involved in materials design and in understanding the complex phenomena causing changes in the microstructure, although it cannot replace experiments. Cai et al. [5] measured internal friction in nanocrystalline aluminum prepared by plasma evaporation and compaction as a function of temperature with five frequencies ranging from 0.3 to 3.1 Hz. They observed two relaxation peaks around 180 ◦ C. Numerical modeling was used by Wang et al. to investigate the damping behaviors of Al/SiC particulate-reinforced metal–matrix composites (PMMCs) at room temperature. Through the cell method and finite-element method, the influences of particulate shape and distribution on the damping capacity of Al/SiC(p) composite are studied [6]. A new model for calculating the damping capacity of particulate-reinforced metal–matrix composites (PMMCs) is proposed by Gu et al. [7]. Their model based on the assumption that the energy loss mainly results from the anelasticity of the particulate and matrix and the micro-plasticity of the matrix under small strain amplitude. Finite element method with a multi-particle model has been adopted. The results showed that the energy loss in the loading direction can represent the total energies consumed in the composites. Trojanová et al. [8] measured strain amplitude dependence of the logarithmic decrement on microcrystalline and nanocrystalline magnesium unreinforced and reinforced with nanoparticles of Al2 O3 , ZrO2 , and graphite. Measurements were carried out before and after step-by-step isochronal thermal treatment at increasing temperature. Their results showed that a decrease in the grain size and secondary phases increase the strain independent component of the decrement. Microstructure changes due to thermal treatment are responsible for an increase of the strain dependent component of the decrement. In a separate study, Liu et al. [9] were measured the low temperature internal friction of a variety of hydrogenated amorphous, nanocrystalline, polycrystalline, and epitaxial silicon thin films. They observed that with increasing volume of nanocrystallinity, the internal friction increases in contrast to our common expectation. The simulation results are compared with the experimental results, which show that they are consistent with each other in quality. Patel et al. [10] used a finite element based unit cell model with periodic boundary conditions to calculate the effective tan ı of a composite as a function of frequency. Their results demonstrated that the tan ı of the nanocomposite manifests important information about the extent and properties of the interphase region that
surrounds each nanoparticle and concluded that measurements of this parameter could provide a simple yet useful experimental tool to understand more about the interphase. However, the measurement of this parameter may prone to error when severe agglomeration or wide size distribution of the nanoparticles are present. Literature review shows that the finite element method is a useful numerical tool capable of predicting the damping properties of metal matrix composites. Research findings of Xu and Schamauder [11], Wang et al. [6,12–13] and Gu et al. [7,14] have been successful in studying the plastic energy dissipation in metal–matrix composites at high strain amplitudes. Although, there have been some progresses in the development of method to estimate the nonlinear elasto-plastic behavior of nanocomposites [15–20], the energy dissipation behaviors of the nanocomposites have not been involved in any of these investigations. In this paper, the finite element method is employed to predict the energy dissipation of nanocomposite material under cyclic loading. A model was developed and then validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposites. Finally, the influence of the grain size and reinforcement volume fraction on the energy dissipation was also calculated by the developed finite element model. 2. Theoretical background 2.1. The model for nanocomposite The nanocomposite contains nanocrystalline (NC) Al–Zn as grain and Al2 O3 nanoparticles as reinforcement. Considering reinforcement as an elastic material, grain and boundary phase are modeled by following assumptions. NC materials are structurally characterized by a large volume fraction of grain boundaries, which may significantly alter their physical, mechanical, and chemical properties in comparison with conventional coarse-grained polycrystalline materials. As the grain size decreases, an increasing fraction of atoms can be ascribed to the grain boundaries. Two types of atoms in the NC structures can be recognized: crystal atoms and boundary atoms. Based on the above model for NC structures, a thickness is assigned for the grain boundary zone in the micrograph which is shown in Fig. 1a. In this figure, the white grains are the reinforcement and the colored grains are representative of the matrix. The difference in orientation of matrix grains is shown in the picture by different degree of grayness. To construct the composite model which is shown in Fig. 1b, some idealization on the shape and arrangement of grains was carried out. The unit cell model considers the harmonic repeating part of multi-cell model. To construct the unit cell model which is shown Fig. 1c, some simplification on the crystallographic orientations of real microstructure and boundary phase arrangement was carried out. For the crystal phase, the elasto-plastic behavior taking grain size into account by Hall–Pitch relation was used, i.e. p
(g)
p
e = y + hg εe (g)
y
(1)
= y∞ + kd−1/2
(2)
p
p
In Eq. (1), e is von Mises’ effective plastic stress and εe is effective plastic strain. The constant hg is the coefficient of hardening. (g)
Eq. (2) relates yield stress (y ) with the grain size by Hall-Pitch relation. In this equation k is a material constant, and d is a grain size. As a general estimation, the boundary thickness is considered independent of grain size with a constant size. As the grain size
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Fig. 1. Derivation of composite model from micrograph.
decreases, the number of grains which take place in a specific area of model increases. Thus, boundary becomes proportionally larger. Based on molecular dynamic (MD) simulation of NC material, Schwaiger et al. [21] suggested a value of 2.5–3.5 nm for boundary phase thickness. The value of 2.5 nm is taken in this study. For the boundary, a quasi-amorphous behavior was assumed. The quasi-amorphous behavior of a metal was modeled using an elastic perfectly plastic curve. In this study, we assume the strength of the boundary phase to increase up to an upper limit and then to stay at that level. The upper limit is set to be the strength of the material in the amorphous state, i.e. am . The stress–strain relation for amorphous materials is considered as ideal plastic behavior, and the assumption of ideal plasticity for the amorphous state is stated to be physically reasonable [17]. We, therefore, model the upper limit in the plastic response of the boundary phase as perfect plasticity. It is generally accepted that the yield stress of amorphous alloys ranges from Eam /50 to Eam /80, while the elastic modulus of the amorphous state, Eam , is approximately 60–75% of that of the corresponding equilibrium crystalline alloy [18].
Fig. 2. The technique used for calculation of W and W.
2.2. Mathematical model for damping capacity
3. Numerical predictions
The energy dissipation ability of the composite material is evaluated by the ratio of the dissipated work by the material W to the maximum of elastic energy stored in the cyclic loading process, and we use to represent the ratio:
Simulation is carried out for nanocrystalline Al–Zn and Al2 O3 as reinforcement. Nanocrystal grain size is approximated to be 40 nm and the nanocomposite contains 14% volume fraction of reinforcement. Fig. 3, depicts the boundary conditions which are applied for modeling of the tensile harmonic loading. The graph clearly depicts the grain, the boundary phase and reinforcement. The boundary conditions required to simulate the damping capacity are assumed as follows:
=
W W
(3)
where W can be written as
W =
dε
(4)
which is the area surrounded by the cyclic stress–strain curve. W is usually expressed as follows:
• For y = 0 and x = 0 surface, the symmetric boundary condition is used. The boundary condition limits displacement in normal direction.
ε0
W=
dε
(5)
0
where ε0 is the maximum strain during the loading process (Fig. 2) while and ε are the macroscopic average stress and strain of the model (Fig. 1). The loading process is quasi-static because what we study here is elasto-plastic deformation. In scientific investigations, the inverse quality factor Q−1 , is widely used to measure the damping capacity of material, which has the relationship with as Q −1 =
2
(6)
With the conception of the energy dissipation and the equations, it is possible to simulate the damping capacity Q−1 , of a material by calculating the area of the hysteresis loop [6–7,11–14].
Fig. 3. The finite element model of the nanocomposite.
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Table 1 Material properties used for simulation [14,20].
E (GPa)
y∞
(MPa) √ k (GPa nm) hg (GPa)
Grain
Boundary phase
Al2 O3 reinforcement
68.3 0.3 58 1.265 2.08
50 0.3 650 – –
380 0.22 – – –
• For the right surface, all nodes on the surface have the same displacement in the direction normal to the surface (i.e., the surface remains un-deformed). This constraint is important in calculations of the unit cell model to avoid necking at the specimen; if this happens, the unit cell model loses its validity. • For the top surface, the surface is loaded by incrementally displacing all nodes lying on the plane in the y-direction. The load is applied along axis y on the top plane of the model and characterized by strain amplitude. The loading process is quasi-static because we focus on elastic-plastic deformation. The set of parameters for simulation is tabulated in Table 1. The damping capacity versus strain amplitude is shown in Fig. 4. The damping curve is made up of a strain amplitude-independent part and a strain amplitude-dependent part. The former reflects the contribution from the an-elasticity of constituents, while the latter mainly corresponds to the plasticity. The critical strain amplitude means the onset of micro-plastic deformation, which depends strongly on the stress state within the material. Stress concentration at the material makes the micro-plastic deformation of grain much easier. These results can be interpreted as follows. In nanocomposites, there are three steps during the deformation process. In the first step, the grain and the boundary phase are elastic and stress concentration due to property mismatch causes damping. In the second step, the grains go to the plastic zone but boundaries remain elastic. Here plastic deformation in grains and stress concentration in grain boundaries cause damping. In the third step, both grain and boundary phase go to the plastic zone and their plastic deformation causes damping. 4. Experimental procedure 4.1. Materials and methods for production of Al–Zn/˛-Al2 O3 nanocompsite A mixture of commercial aluminum (99.7% purity and particle size of 50–70 m) and 15.8 wt.% ZnO powders (99.9% purity and particle size of 250 nm) was milled in planetary ball mill in order to produce Al–14 wt.%Zn/5 vol%Al2 O3 nanocomposite. The Mechanical Alloying (MA) was executed with a rotation speed of 600 rpm for predetermined hours without interruption, and the ball-powder mass ratio was 15:1. A mixture of aluminum and 14 wt.% zinc pow-
Fig. 4. Damping capacity curve versus strain amplitude in nanocomposite with grain size 40 nm.
Fig. 5. Transmission electron microscopy (TEM) of Al–Zn/␣-Al2 O3 nanocomposite.
ders (99.7% purity and particle size of 10–20 m) was also milled with a planetary ball mill in same condition. The produced powders was filled in a uniaxial die made of X40CrMoV51 (AISI H13) hot work tool steel. Then, the specimen was heated to 400 and 500 ◦ C and pressed at constant pressures of 400 MPa. The duration of hot pressing was 20 min. In order to avoid pores formation, the pressure on each specimen was not released until the specimen was cooled down to 300 ◦ C [22–23]. Transmission electron microscopy (TEM) picture of the nanocomposite is shown in Fig. 5. For our case study, the powder was poured into a cylindrical die and pressed regarding to the above mentioned procedure to create a cylinder with diameter of 50 mm and height of 15 mm. The object was then used in order to find out its modal and damping properties which are described in the next section. Detailed description of powder processing and bulk material preparation can be found in references [22] and [23], respectively. 4.2. Modal testing and analysis The damping factor of the specimen was determined by use of modal testing [24–26]. The measurement principle consists of recording the free vibrations of the specimen excited by tapping it with an appropriate hammer, as shown in Fig. 6. The amplitude decay as a function of time and the vibration modes were detected by an acquisition data system from B&K Company and recorded using ICATS software [27] and also by analytical relations. The test specimen was located on soft foam and tested in a free-free condition. The experimental data were obtained via hammer testing using a B&K Pulse analyzer connected to a PC. One B&K piezoelectric accelerometer (Endevco. 2222c) was attached to the structure by using wax to measure the amplitude decay of the response (Fig. 6). The test parameters were: analyses range of 0–20,000 Hz; acquisition time of 320 ms; rectangular observation window and frequency resolution of 3.125 Hz. Following the testing procedure, two types of curves were obtained: damping free vibration and frequency response function profiles. A total of three FRFs were measured and saved in appropriate format in the computers.
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Fig. 6. The experimental set-up for modal testing of the specimen.
The vibration test gives the free vibration damping decay and the frequency response function (FRF), simultaneously as a result. Considering a linear system of a single degree of freedom, the FRF response is the decomposition of the natural frequencies of a structure or specimen, which corresponds to a typical fingerprint identity of the vibration modes. The number of vibration peak frequencies (vibration modes) and the shape of the FRF response are a direct result of the material damping. In the second stage, the experimental FRF data were analyzed using a global multi-FRF analysis method and the results were also checked using available analytical relations [28]. The global multiFRF technique is based on a complex singular value decomposition of a system matrix expressed in terms of measured FRFs, and then on a complex eigensolution which extracts the required modal properties [27–28]. The data were analyzed by applying three runs on each FRF. Typical measured FRF is plotted in Fig. 7. An examination of the response curves shows that the shape of these curves is controlled by the amount of damping present in the system, in particular the bandwidth, that is the difference between two frequencies corresponding to the same response amplitude is related to the damping of the system. In evaluation of damping, it √ is convenient to measure the bandwidth at 1/ 2 of peak amplitude (Fig. 8). The frequencies corresponding in this bandwidth ω1 and ω2 are referred to as half-power points. It can be showed that the damping can be estimated easily by the following relation [24]: Q −1 =
ω2 − ω1 ωr
(7)
where Q−1 is the damping factor and ωr is the peak frequency in the response curves (natural frequency). In order to check the validity of the global multi-FRF method, the damping factor was also
Fig. 7. The measured frequency response function from excitation of the specimen.
Fig. 8. The half-power point technique for modal analysis.
calculated from the half-power rule (i.e. Eq. (7)). It can be seen from Table 2 that the error depends on the analysis method used for extracting the parameters from the measured data. The modal analysis will offer more accurate results as more complicated technique was used for parameter estimation. However, the difference between measured and calculated damping factor is at an acceptable level. 5. Parametric study It was mentioned previously that modeling can be a useful tool in predicting material behavior and effectively reducing the number of experimental trials involved in materials design. Having constructed and validated the finite element model, it was decided to carry out a parametric study in order to find out the effects of different parameters on the damping capacity of the nanocomposite. Damping capacity versus strain amplitude for different volume fraction of reinforcement is shown in Fig. 9. It can be observed that
Fig. 9. Damping capacity versus strain amplitude for different size of reinforcement: (a) 20 nm and (b) 30 nm.
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Table 2 Measure and estimated results for the specimen at low strain rate. Estimated parameter
Damping factor Q−1 Natural frequency (Hz) *
Experiment
Finite element method
Modal analysis
Eq. (7)
0.1177 (Error* 5.8%) 13750
0.1095 (Error = 12.4%) 13700
0.125 (Fig. 4) N/A
FEM Error (%) = Measured × 100. FEM
Finally, the influence of the grain size and reinforcement volume fraction on the energy dissipation was calculated by finite element model. It was found that by increasing the reinforcement volume fraction damping capacity versus strain amplitude would increase the damping capacity of nanocomposite. The results also indicate that decreasing the grain size enhance the damping in the nanocomposites. References
Fig. 10. Damping capacity versus strain amplitude for different grain size: (a) 30 nm; (b) 40 nm; (c) 50 nm; and (d) 100 nm.
as the reinforcement volume fraction increases, the curve shifts upward especially in larger strain amplitude. The reason is mainly due to more stress concentration in the grain and boundary phase. Damping capacity versus strain amplitude for different grain size of nanocrystalline matrix is shown in Fig. 10. It also can be observed that smaller grain size causes a downward shifting in the curve. The reason is mainly due to hardening of material and therefore reducing its damping by grain refinement. 6. Conclusions In the present study, the finite element method is employed to estimate the energy dissipation of nanocomposite material under cyclic loading. Materials and methods for production of Al–Zn/␣-Al2 O3 nanocompsite were described. Several nanocomposite cylinders were constructed and used for verifying the proposed model. The model was validated by comparing the numerical results against an impact based suspended cylinder experiment conducted at low strain amplitude on Al–Zn/␣-Al2 O3 of nanocomposites. The good agreement between the predicted and measured damping capacity indicate the capability of the proposed model in estimating the damping capacity of reinforced nanocomposite.
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