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A standard experimental method for determining the material length scale based on modified couple stress theory
Accepted Manuscript
A standard experimental method for determining the material length scale based on modified couple stress theory Zhenkun Li , Yuming He , Jian Lei , Song Guo , Dabiao Liu , Lin Wang PII: DOI: Reference:
S0020-7403(18)30087-0 10.1016/j.ijmecsci.2018.03.035 MS 4245
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 January 2018 12 March 2018 26 March 2018
Please cite this article as: Zhenkun Li , Yuming He , Jian Lei , Song Guo , Dabiao Liu , Lin Wang , A standard experimental method for determining the material length scale based on modified couple stress theory, International Journal of Mechanical Sciences (2018), doi: 10.1016/j.ijmecsci.2018.03.035
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Highlights
A standard experimental method for determining the intrinsic material length scale of modified couple stress theory is proposed.
Material length scale parameter of modified couple stress theory is determined for two metallic materials: copper and titanium.
The strength of size effects are compared between metallic and non-metallic materials. The influences of material properties (Young’s modulus and mass density) on size
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effects are discussed.
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A standard experimental method for determining the material length scale based on modified couple stress theory Zhenkun Li1,2 Yuming He1,2,* ,Jian Lei1,2,*, Song Guo1,2 , Dabiao Liu1,2 and Lin Wang1,2 a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
b
Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China
*
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Corresponding author at Huazhong University of Science and Technology, Wuhan 430074, China E-mail address: [email protected] (Y. He), [email protected] (J. Lei)
Abstract: A standard experimental method for determining the intrinsic material length scale
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of modified couple stress theory is proposed based on a non-contact laser Doppler vibration measurement system. The first-order resonant frequencies of two different cantilever microbeams made of copper and titanium with thickness ranging from 15 to 2 μm are measured, respectively. The elastic size effects are obviously observed from the experimental results. The length scale parameters are obtained by fitting the experiment data with the
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modified couple stress theory. Multiply verified by different kind of materials, we considered
couple stress theory.
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this method a standard and valid way for determining the material length scales of modified
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Keywords: Size effect; Material length scale; Modified couple stress theory; Microbeams;
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Vibration measurement
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1 Introduction Small-scale structures such as microbeams and microplates, due to their sensitivity and excellent performance in small sizes, have been widely used in micro/nano electro mechanical systems (MEMS/NEMS). The common usage includes micro-resonators [1], sensors and actuators[2], atomic force microscopes (AFMs) [3], and micro-switches [4], etc. Among these
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small-scale structures, one of the characteristic dimensions like thickness is in the micro- or nano-meter scale [5-13].
It has been vastly confirmed that the mechanical behaviors of small-scale structures are size-dependent. In the field of plasticity, Fleck et al. [5] firstly found that the normalized torque increases dramatically with the wire diameter decreasing from 170 to 20 µm in the torsion tests of polycrystalline Cu wires. Thereafter, Liu et al. [10, 14] also discovered the
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significant size effects at initial yielding and plastic flow stress in the torsion test of polycrystalline Cu wires with diameters from 18 to 105 µm. Similar phenomena were also confirmed in other loading conditions, for instance, foil bending [6], and micro- and nano-indentation [15, 16]. In the elastic range, the size-dependent effect of polymers were found in the scale of nanometer [2, 17-20]. On the order of microns, only a few experiments on size-dependent elasticity can be found [21, 22]. For example, Lam et al. [21] developed a
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strain gradient elasticity theory and conducted bending tests on epoxy cantilever microbeams. They found that the normalized beam rigidity exhibited an inverse squared dependence on the
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thickness of microbeam, i.e. the normalized bending rigidity increases about 2.4 times as the thickness of the microbeam reduced from 115 to 20 µm. To account for the size effect observed in micro and nano scales, researchers have
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developed various higher order continuum theories involving additional material length scales. In chronological order there are the classical couple stress theory [23, 24], nonlocal elasticity
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theory [25], surface elasticity theory [26, 27], modified couple stress theory (MCST) [28], strain gradient theory (SGT) [5, 7], and nonlocal strain gradient theory[29, 30]. Taking SGT into account, it contains three length scale parameters l0 , l1 , l2 , corresponding to the
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dilatation gradient vector, the deviatoric stretch gradient tensor and the curvature tensor, respectively. As a special case of SGT, the MCST only contains one material length scale l which is related to the curvature tensor. The length scales introduced in these theories are generally considered to be inherent material parameters. Along with these theories, a large deal of theoretical and numerical studies on mechanical responses of small-scale structures have been performed in the past two decades: static and dynamic behaviors are investigated employing higher-order theories in microbeams [19, 31-39], microplates and shells [20, 40-46] as well as tubes and pipes [47-53]. For instance, Chen et al. [33] developed a composite laminated Timoshenko beam (CLTB) model based on MCST and the size effects of the
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natural frequencies within the free vibration of CLTB was captured. Shaat et al. [40] conducted bending analysis for Size-dependent Kirchhoff nano-plates incorporating surface effects based on a MCST. More recently, Fang et al. [36] studied the free vibration of rotating functionally graded (FG) microbeams via establishing a size-dependent three-dimensional dynamic model incorporating MCST and Euler–Bernoulli beam theory. However, due to the difficulty in conducting the experiments at small scale, only a few attempts have been devoted to the measurement of the length scale parameters. According to
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the experiments performed by Lam et al. [21], Park et al. [31] and Kahrobaiyan et al. [54] have calculated the value of the scale parameter in MCST and SGT, respectively. Recently, Liebold et al. [22] performed AFM experiments with microbeams made of epoxy and polymer SU-8, the corresponding material length scale in SGT has been determined.
It is noted that most of the experiments on the size dependent elasticity at the micron scale, e.g. static AFM bending tests [21, 22] and micro/nano- indentation tests [15, 16], are
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confined to non-metallic materials like silicon or epoxy. More recently, Lei et al.[55] in our group carried out an dynamic experiment on nickel cantilever microbeams based on the laser Doppler vibrometer (LDV) for the first time. The resonant frequencies of the cantilever microbeams were measured, which indicates that the dimensionless natural frequency increases to about 2.1 times with the beam thickness decreasing from 15 to 2.1 µm. By fitting to the experimental data, the material length scale based on the strain gradient elasticity
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theory (SGT) and the modified couple stress theory (MCST) were calculated, respectively. It is noticeable that the measured length scale parameters are significantly different for
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metal [55] and non-metal materials [21, 22, 31, 54]. For instance, the length scale parameter of metallic nickel (Ni) corresponding to modified couple stress theory [55] is 1.553 µm while it is of 17.6 µm [21, 31] and 2.5 µm [22] for epoxy and SU-8, respectively. Thus what the
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length scale parameters in other metallic materials would be? To that end, microbeams made of two other different metallic materials, i.e. titanium (Ti) and copper (Cu), are adopted in the
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current experiments to determine the length scale parameter. A strong dependence of the dimensionless first-order resonant frequency of microbeams on the beam thickness is observed. By fitting to the experimental data, length scale parameter of MCST is obtained for
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each material. It is shown that the method developed here can be considered a standard and reliable method for determining the material length scale for modified couple stress theory.
2 Experimental details To determine the material length scale by the aid of experiment, the thickness of metallic materials should be chosen at the order of micrometer based on the knowledge of previous experiments [55]. Commercially available metallic foils that can satisfy the experiment condition are Ti and Cu (99.999% purity). The nominal thickness of Ti foils is 2, 5, 10 and 15
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µm, while the nominal thicknesses of Cu foil thinner than 10 µm are absent. To obtain much thinner Cu foils, the electropolishing technique [56] is applied to reduce the thickness as well as smooth the foil surface. The device for electropolishing is depicted in Fig. 2. The anode and cathode of the device are separated by an insulative polytetrafluoroethylene (PTFE) frame with thickness of 10 mm. Four stainless steel rods embedded in each corner of the PTFE frame serve as the anode (see Fig. 2.). Cu strips with length of about 110 mm to be electropolished are clamped between the crossbeams connected
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to anode rods. To produce a uniformly distributed electric field, a stainless steel plate at the bottom of the device connected with a stainless steel rod constitutes the cathode.
The electrolyte solution is a mixture of ethanol ( C2 H5 OH ) and phosphoric acid ( H3 PO 4 ) with volume ratio of 1:9 [56]. During electropolishing, the Cu strips are immersed in the solution, while the other sections of the five rods are above the electrolyte solution level and connected to a DC power supply. The DC power supply works in a constant current mode of 3.37 A. The
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positive port connects to all four anodic rods. The negative terminal is attached to the other rod in the cathode plate. The surfaces of the Cu strips are dissolved in the electrolyte to remove oxygen. At the cathode, a reduction reaction occurs, which produces hydrogen and Cu. The initial Cu foil of nominal thickness 10 μm are electropolished at room temperature. In this way, Cu foils with different thickness can be obtained under different electropolishing time.
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The size of a cantilever microbeam is intentionally designed as 1 mm wide and 5 mm long [55]. Too long a cantilever microbeam can easily lead to buckling while too short is not
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suitable for a beam theory and also cause difficulties in assembling. Firstly Ti and Cu foils of each thickness were cut into rectangular strips with width of 1 mm and length of 10 mm by a sharp knife blade. The samples are then annealed at 300℃ for 5 hours in a vacuum furnace to
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eliminate residual stress. Without this process, the resultant experimental data would be too scattered to be analyzed due to the residual stress induced in the former step. Finally one end
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of each strip is glued to a rigid plate with 5 mm, while the other end extends out of the plate edge. Thus cantilever microbeams with width of 1 mm and length of 5 mm were made and subsequently mounted on the three-dimensional translation stage.
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The dynamic vibration measurement system consisting of a laser Doppler vibrometer
(LDV), a direct digital frequency synthesis (DDS) signal generator with minimum resolution of 0.01 Hz, a loudspeaker with power of 2 W, a computer, and a three-dimensional (3D) translation stage is illustrated in Fig. 1. The LDV probe with a laser beam diameter of 50 μm at the focused point is used for vibration measurement of the microbeam. And then the detected vibration response of the cantilever microbeam is displayed on the computer connected to the LDV. The DDS signal generator here can generate a continuous sinusoidal acoustic wave to excite the cantilever microbeam by the aid of loudspeaker. This kind of non-contact excitation excludes the influence of additional mass.
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Before the experiment, the distance and angular between the laser probe and the cantilever microbeam is elaborately adjusted to ensure the laser beam is focused on the free end of the microbeam. The cantilever microbeam is placed uprightly on the 3-D translation stage which can make the laser beam transmitted from the LDV be reflected the same way back to the laser probe. The vibration response of the cantilever microbeam is then detected and displayed on the computer connected to the LDV. In order to obtain the resonant frequency of the cantilever microbeam, the excitation frequency of the sinusoidal signal starts
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from a relatively low frequency less than the natural frequency and is increased gradually. Each time after increase, wait for 5 seconds to allow the response of the cantilever microbeam to stay stable, and then the amplitude of the vibration is recorded. When the excitation frequency approaches the resonant frequency, adjust the excitation frequency at the step of 0.1 Hz for sake of accuracy. The process continues until the recorded amplitude no longer increases, which indicates the excitation frequency has surpassed the resonant frequency.
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Thus a complete frequency-response curve near the resonant frequency is obtained. From the frequency-response curve, one can read the value of resonant frequency, and the error is within 0.1 Hz. Some typical frequency-response curves of Ti and Cu cantilever microbeams in different thicknesses are given in Fig. 4. It can be seen that the maximum magnitude of response amplitude is less than the corresponding thickness, also the frequency-response curve is symmetric near the peak of the curve. Conclusion can be made that vibration of the
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microbeams is in linear elastic range. Each specimen is measured at least three times and the experimental data are averaged, and for every single thickness, at least five specimens are
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prepared. Details of the experiment can be found in the previous work [55]. To reduce experimental error, the exact length and width of the cantilever microbeams are examined in an optical microscope to ensure the tolerance is within 0.05 mm before the
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tests. Furthermore, after the experiments, the actual thickness of Ti and Cu samples are measured through cutting a square hole vertically in the center of specimens with focused ion
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beam (FIB), as shown in Fig. 3(a) and (b). Besides, the surface roughness of each sample is tested in a laser scanning confocal microscopy (LSCM). A typical surface morphology is depicted in Fig. 3(c) and (d). The full data of thickness and roughness are listed in Table 1.
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One can see that the roughness of each sample is small enough to perform the experiment.
3 Length scale parameters fitting To obtain the length scale parameter, the equation of motion of size dependent microbeams are derived [57], i.e.
AW ( EI Al 2 )W (4) 0
(1)
Where l is the material length scale for modified couple stress theory, EI is the bending
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(2)
Substituting Eq. (2) into (1) and separating variables leads to
cos sn L cosh sn L +1=0
(n 1,2,3,...)
sn L 1.875， 4.694, 7.855, ... Then finally yields
EI Al 2 （sn L）2 , 4 AL
(n 1, 2, 3, )
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n
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where
(3)
(4)
(5)
According to the classical Bernoulli-Euler beam theory [31], the natural frequency of a cantilever beam is given by
cn
EI 2 （s L） , AL4 n
(n 1, 2, 3, )
(6)
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Where subscript c in cn stands for classic solution.
Combining Eq. (5) and Eq.(6), the dimensionless frequency is obtained
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1 l 1 6( ) 2 c1 h
(7)
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Here, for simplicity, the Poisson’s effect is neglected [57], i.e. 0 .
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Effects of Poisson’s ratio and plain strain state For a slender beam with a large aspect ratio, the Poisson effect is secondary and may be
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neglected to facilitate the formulation of a simple beam theory [31], as demonstrated in the above section. For beams with relatively small aspect ratio, taking into account the influence of Poisson effect, Eq. (7) is rewritten as
1 6 l 2 1 ( ) c1 1+ h
(8)
Where substitution
is applied in Eq. (5).
E 2(1 )
(9)
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On the other hand, when the thickness of the thin beam is much smaller than its length and width, e.g. b/h>5, the displacement along the width direction is ignored hence plane strain state should be considered. In this case the bending rigidity EI in Eq. (5), (6) is transformed to EI , consequently the dimensionless frequency yields (1 2 )
(10)
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1 l 1 6(1- )( )2 c1 h
By fitting to the experimental data with Eq. (7) utilizing the least squares method, one can obtain the material length scale in MCST model, which gives l=1.442 μm for Cu and l=0.775 μm for Ti. In the case that Poisson’s effect is considered, the material length scale parameters are fitted via Eq. (8) as l=1.573 μm and l=0.877 μm for Cu and Ti respectively;
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And in the present case plain strain state is satisfied, from Eq. (10) one can obtain the material length scale parameter as l=1.661 μm for Cu and l=0.922 μm for Ti. In the last two situations the material length scale parameter is around 10% more than the corresponding case where Poisson’s ratio is ignored.
Also, the material length scale parameters for SGT can be solved based on the resultant experimental data if one assumes that all the three parameters are the same, i.e. l0 l1 l2 l
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. Of course it’s only for discussion, as pointed out by Lam [21]. The governing equation are determined as follow [55]:
4W 6W 2W K A 0 x 4 x 6 t 2
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S
where
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4 8 K I 2l02 l12 , S EI 2 Al02 Al12 Al22 5 15
(11)
(12)
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Taking advantage of the technique in literature [55], where differential quadrature
method (DQM) is applied to solve the governing equation and the method of minimization of
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least squares of the error between the experimental data and the SGT results is used to fit in the curve. The length scale parameters in SGT are obtained as 0.631 µm for Cu and 0.353 µm for Ti, respectively. Details of the calculation procedure are referred to literature [55]. It’s noted that there are higher-order boundary conditions for SGT however classic boundary conditions are employed in the above calculation for comparison to the result in literature [55]. Considering non-classic boundary conditions [54]and repeating the calculation procedure, the length scale parameters in SGT are as 0.604 µm for Cu and 0.311 µm for Ti. The errors are respectively around 10% and 5% comparing to the results obtained by classic boundary conditions. The dimensionless bending rigidity (the ratio of bending rigidity) is defined as
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EI Al 2 ( 1 )2 EI c1
(13)
The dimensionless frequency and the dimensionless bending rigidity with different thicknesses are shown in Fig. 5. As seen, the dimensionless frequency of Cu microbeams increases to as high as 1.6 times with decreasing the thickness of cantilever microbeam from 13.12 to 2.79 µm, while the dimensionless bending rigidity increases to about 2.6 times. Similarly, the dimensionless frequency and the dimensionless bending rigidity of Ti
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microbeams are increased to 1.38 times and 1.9 times when the microbeam thickness is decreased from 14.69 to 2.02µm. It indicates the inconsistency between the experimental natural frequencies and the classical solutions increases with the decreasing thickness, hence there exists size effects.
The fitted dimensionless frequencies curves based on the SGT, MCST and classical theory (l = 0) of Cu and Ti are displayed in Fig. 6. One can see that the SGT and the MCST
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give nearly the same curve, and that the experimental data is in good agreement with that predicted by higher order continuum theories.
The fitted elastic length scale parameters are shown in Table 2. For comparison, results obtained in the same condition as in the literature [55] are used. One can see that, compared to the material of epoxy, the material length scale of metallic materials are much smaller, about
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one-tenth [21, 31] and one-sixth [22] of each epoxy material. It reveals that the intrinsic structure between the metallic materials and the epoxy is different. The fitted curves of
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metallic (in the same condition) and non-metallic materials based on the MCST are depicted in Fig. 7. Since the SGT model gives nearly the same curve as the MCST, as seen in Fig. 6, the fitted curves based on SGT are omitted. One can see that the higher the length scale
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parameter is, the stronger the size effect becomes whether it is metallic or non-metallic materials. Moreover, there is an interesting observation from Table 2 that the value of length scale parameters for Cu and Ni are similar to each other when their densities are almost the
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same, although the Young’s modulus of Cu is merely half of that of Ni. However, Cu and Ti are similar in Young’s modulus but the density of Cu is twice of that of Ti. Dramatically, the
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length scale parameter in Cu is also nearly twice of Ti. This observation suits for length scale parameters based on both MCST and SGT. It seems mass density has greater influence on the length scale parameter than Young’s modulus. To further understand the relationship between length scale parameter and the material properties requires more experimental data in the near future.
4 Conclusion A standard and reliable method for determining the material length scale based on
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modified couple stress theory (MCST) is proposed in the paper. The method has been successfully applied to two metallic materials, Cu and Ti. As a result, evident elastic size effects are observed. By fitting to the experimental data, the length scale parameter of MCST is determined. Comparing the results of SGT and MCST, we find the value of length scale parameter in MCST is almost twice of that in SGT. The comparison between the epoxys and the metallic foils indicates a huge difference in the length scale parameters for these different materials.
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The reason could be related to the intrinsic material structures. Comparison between metals indicates that nickel foil has the strongest size effect, while Ti has the weakest size effect. And furthermore mass density seems to have greater influence on the length scale parameter than Young’s modulus.
Finally, compared to the static bending experiments, the method developed here is more convenient and economical to implement, thus can be used as a standard method for
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determining the material length scale in modified couple stress theory. The method will be useful for theoretical and numerical simulation of micro-structures and is of great importance for the design of MEMS/NEMS.
Acknowledgments
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This work was financially supported by the National Natural Science Foundation of
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China (Nos. 11772138, 11472114, 11572133, 11702103). DL also thanks the financial support of the Young Elite Scientist Sponsorship Program by CAST (No. 2016QNRC001).
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[47] Wang L. Size-dependent vibration characteristics of fluid-conveying microtubes. Journal of Fluids and Structures, 2010; 26(4): 675-84.
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[48] Zhou X,Wang L Vibration and stability of micro-scale cylindrical shells conveying fluid based on modified couple stress theory. Micro & Nano Letters, 2012. 7, 679-84. [49] Zhou X, Wang L, Qin P. Free Vibration of Micro- and Nano-Shells Based on Modified Couple Stress Theory. Journal of Computational and Theoretical Nanoscience, 2012; 9(6): 814-18. [50] Akgöz B,Civalek O. Buckling analysis of linearly tapered micro-Columns based on strain gradient elasticity. Structural Engineering and Mechanics, 2013; 48(2): 195-205.
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[51] Hu K, Wang YK, Dai HL, Wang L, Qian Q. Nonlinear and chaotic vibrations of cantilevered micropipes conveying fluid based on modified couple stress theory. Int J Eng Sci, 2016;
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105(Supplement C): 93-107.
[52] Akgöz B,Civalek Ö. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica, 2016; 119: 1-12.
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[53] Civalek Ö,Demir C. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 2016; 289:
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335-52.
[54] Kahrobaiyan M, Asghari M, Ahmadian M. Strain gradient beam element. Finite Elem Anal Des, 2013; 68: 63-75.
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[55] Lei J, He Y, Guo S, Li Z, Liu D. Size-dependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity. Aip Adv, 2016; 6(10): 105202.
[56] Guo S, He Y, Lei J, Li Z, Liu D. Individual strain gradient effect on torsional strength of electropolished microscale copper wires. Scripta Mater, 2017; 130: 124-27.
[57] Kong S, Zhou S, Nie Z, Wang K. The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int J Eng Sci, 2008; 46(5): 427-37.
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Fig. 1. Schematic diagram of vibration measurement system.
Fig. 2. Schematic diagram of the apparatus for electropolishing.
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Fig. 3. (a) and (b):Typical images of FIB thickness measurement of Cu and Ti;
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(c) and (d):Typical surface morphology of Cu and Ti(in laser color picture mode).
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Fig. 4. Amplitude-frequency curves of: (a) Ti, (b) Cu.
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Fig. 5. Dimensionless frequency for: (a) Ti, (b) Cu; Dimensionless bending rigidity for: (c) Ti, (d) Cu.
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Fig. 6. Mean values of the experimental data, SGT solution, MCST solution and the classical solution of the dimensionless natural frequencies for: (a) Ti, (b) Cu.
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Fig. 7. Size effect comparision of different materials, (a): Ni [55], Cu, Ti; (b): Epoxy [21], Epoxy [22] and SU-8 [22].
Table 1
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Ti
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Cu
Thickness/μm
Ra/nm
14.690
23.7
10.440
60.7
5.410
48.5
2.015
45.3
13.123
45.5
9.093
48.0
6.490
62.5
4.206
35.1
2.790
28.4
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Thickness and roughness of Ti and Cu foils.
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Table 2 Length scale parameters of each material.
Poisson's ratio
ρ/kg/m3
Ni[55]
207
--
8900
Cu
108
0.32
8900
Ti
108
0.31
4505
Epoxy[21]
1.44
--
--
Epoxy[22]
3.93
--
--
1.44
--
--
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SU-8[22]
theory
l/μm
SGT MCST SGT MCST SGT
0.843 1.553 0.631 1.422 0.353
MCST SGT MCST SGT MCST SGT
0.775 11.01 17.60 4.35 7.75 1.39
MCST
2.50
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E/Gpa
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material
A standard experimental method for determining the material length scale based on modified couple stress theory Zhenkun Li , Yuming He , Jian Lei , Song Guo , Dabiao Liu , Lin Wang PII: DOI: Reference:
S0020-7403(18)30087-0 10.1016/j.ijmecsci.2018.03.035 MS 4245
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 January 2018 12 March 2018 26 March 2018
Please cite this article as: Zhenkun Li , Yuming He , Jian Lei , Song Guo , Dabiao Liu , Lin Wang , A standard experimental method for determining the material length scale based on modified couple stress theory, International Journal of Mechanical Sciences (2018), doi: 10.1016/j.ijmecsci.2018.03.035
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Highlights
A standard experimental method for determining the intrinsic material length scale of modified couple stress theory is proposed.
Material length scale parameter of modified couple stress theory is determined for two metallic materials: copper and titanium.
The strength of size effects are compared between metallic and non-metallic materials. The influences of material properties (Young’s modulus and mass density) on size
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effects are discussed.
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A standard experimental method for determining the material length scale based on modified couple stress theory Zhenkun Li1,2 Yuming He1,2,* ,Jian Lei1,2,*, Song Guo1,2 , Dabiao Liu1,2 and Lin Wang1,2 a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
b
Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China
*
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Corresponding author at Huazhong University of Science and Technology, Wuhan 430074, China E-mail address: [email protected] (Y. He), [email protected] (J. Lei)
Abstract: A standard experimental method for determining the intrinsic material length scale
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of modified couple stress theory is proposed based on a non-contact laser Doppler vibration measurement system. The first-order resonant frequencies of two different cantilever microbeams made of copper and titanium with thickness ranging from 15 to 2 μm are measured, respectively. The elastic size effects are obviously observed from the experimental results. The length scale parameters are obtained by fitting the experiment data with the
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modified couple stress theory. Multiply verified by different kind of materials, we considered
couple stress theory.
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this method a standard and valid way for determining the material length scales of modified
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Keywords: Size effect; Material length scale; Modified couple stress theory; Microbeams;
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Vibration measurement
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1 Introduction Small-scale structures such as microbeams and microplates, due to their sensitivity and excellent performance in small sizes, have been widely used in micro/nano electro mechanical systems (MEMS/NEMS). The common usage includes micro-resonators [1], sensors and actuators[2], atomic force microscopes (AFMs) [3], and micro-switches [4], etc. Among these
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small-scale structures, one of the characteristic dimensions like thickness is in the micro- or nano-meter scale [5-13].
It has been vastly confirmed that the mechanical behaviors of small-scale structures are size-dependent. In the field of plasticity, Fleck et al. [5] firstly found that the normalized torque increases dramatically with the wire diameter decreasing from 170 to 20 µm in the torsion tests of polycrystalline Cu wires. Thereafter, Liu et al. [10, 14] also discovered the
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significant size effects at initial yielding and plastic flow stress in the torsion test of polycrystalline Cu wires with diameters from 18 to 105 µm. Similar phenomena were also confirmed in other loading conditions, for instance, foil bending [6], and micro- and nano-indentation [15, 16]. In the elastic range, the size-dependent effect of polymers were found in the scale of nanometer [2, 17-20]. On the order of microns, only a few experiments on size-dependent elasticity can be found [21, 22]. For example, Lam et al. [21] developed a
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strain gradient elasticity theory and conducted bending tests on epoxy cantilever microbeams. They found that the normalized beam rigidity exhibited an inverse squared dependence on the
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thickness of microbeam, i.e. the normalized bending rigidity increases about 2.4 times as the thickness of the microbeam reduced from 115 to 20 µm. To account for the size effect observed in micro and nano scales, researchers have
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developed various higher order continuum theories involving additional material length scales. In chronological order there are the classical couple stress theory [23, 24], nonlocal elasticity
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theory [25], surface elasticity theory [26, 27], modified couple stress theory (MCST) [28], strain gradient theory (SGT) [5, 7], and nonlocal strain gradient theory[29, 30]. Taking SGT into account, it contains three length scale parameters l0 , l1 , l2 , corresponding to the
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dilatation gradient vector, the deviatoric stretch gradient tensor and the curvature tensor, respectively. As a special case of SGT, the MCST only contains one material length scale l which is related to the curvature tensor. The length scales introduced in these theories are generally considered to be inherent material parameters. Along with these theories, a large deal of theoretical and numerical studies on mechanical responses of small-scale structures have been performed in the past two decades: static and dynamic behaviors are investigated employing higher-order theories in microbeams [19, 31-39], microplates and shells [20, 40-46] as well as tubes and pipes [47-53]. For instance, Chen et al. [33] developed a composite laminated Timoshenko beam (CLTB) model based on MCST and the size effects of the
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natural frequencies within the free vibration of CLTB was captured. Shaat et al. [40] conducted bending analysis for Size-dependent Kirchhoff nano-plates incorporating surface effects based on a MCST. More recently, Fang et al. [36] studied the free vibration of rotating functionally graded (FG) microbeams via establishing a size-dependent three-dimensional dynamic model incorporating MCST and Euler–Bernoulli beam theory. However, due to the difficulty in conducting the experiments at small scale, only a few attempts have been devoted to the measurement of the length scale parameters. According to
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the experiments performed by Lam et al. [21], Park et al. [31] and Kahrobaiyan et al. [54] have calculated the value of the scale parameter in MCST and SGT, respectively. Recently, Liebold et al. [22] performed AFM experiments with microbeams made of epoxy and polymer SU-8, the corresponding material length scale in SGT has been determined.
It is noted that most of the experiments on the size dependent elasticity at the micron scale, e.g. static AFM bending tests [21, 22] and micro/nano- indentation tests [15, 16], are
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confined to non-metallic materials like silicon or epoxy. More recently, Lei et al.[55] in our group carried out an dynamic experiment on nickel cantilever microbeams based on the laser Doppler vibrometer (LDV) for the first time. The resonant frequencies of the cantilever microbeams were measured, which indicates that the dimensionless natural frequency increases to about 2.1 times with the beam thickness decreasing from 15 to 2.1 µm. By fitting to the experimental data, the material length scale based on the strain gradient elasticity
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theory (SGT) and the modified couple stress theory (MCST) were calculated, respectively. It is noticeable that the measured length scale parameters are significantly different for
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metal [55] and non-metal materials [21, 22, 31, 54]. For instance, the length scale parameter of metallic nickel (Ni) corresponding to modified couple stress theory [55] is 1.553 µm while it is of 17.6 µm [21, 31] and 2.5 µm [22] for epoxy and SU-8, respectively. Thus what the
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length scale parameters in other metallic materials would be? To that end, microbeams made of two other different metallic materials, i.e. titanium (Ti) and copper (Cu), are adopted in the
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current experiments to determine the length scale parameter. A strong dependence of the dimensionless first-order resonant frequency of microbeams on the beam thickness is observed. By fitting to the experimental data, length scale parameter of MCST is obtained for
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each material. It is shown that the method developed here can be considered a standard and reliable method for determining the material length scale for modified couple stress theory.
2 Experimental details To determine the material length scale by the aid of experiment, the thickness of metallic materials should be chosen at the order of micrometer based on the knowledge of previous experiments [55]. Commercially available metallic foils that can satisfy the experiment condition are Ti and Cu (99.999% purity). The nominal thickness of Ti foils is 2, 5, 10 and 15
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µm, while the nominal thicknesses of Cu foil thinner than 10 µm are absent. To obtain much thinner Cu foils, the electropolishing technique [56] is applied to reduce the thickness as well as smooth the foil surface. The device for electropolishing is depicted in Fig. 2. The anode and cathode of the device are separated by an insulative polytetrafluoroethylene (PTFE) frame with thickness of 10 mm. Four stainless steel rods embedded in each corner of the PTFE frame serve as the anode (see Fig. 2.). Cu strips with length of about 110 mm to be electropolished are clamped between the crossbeams connected
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to anode rods. To produce a uniformly distributed electric field, a stainless steel plate at the bottom of the device connected with a stainless steel rod constitutes the cathode.
The electrolyte solution is a mixture of ethanol ( C2 H5 OH ) and phosphoric acid ( H3 PO 4 ) with volume ratio of 1:9 [56]. During electropolishing, the Cu strips are immersed in the solution, while the other sections of the five rods are above the electrolyte solution level and connected to a DC power supply. The DC power supply works in a constant current mode of 3.37 A. The
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positive port connects to all four anodic rods. The negative terminal is attached to the other rod in the cathode plate. The surfaces of the Cu strips are dissolved in the electrolyte to remove oxygen. At the cathode, a reduction reaction occurs, which produces hydrogen and Cu. The initial Cu foil of nominal thickness 10 μm are electropolished at room temperature. In this way, Cu foils with different thickness can be obtained under different electropolishing time.
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The size of a cantilever microbeam is intentionally designed as 1 mm wide and 5 mm long [55]. Too long a cantilever microbeam can easily lead to buckling while too short is not
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suitable for a beam theory and also cause difficulties in assembling. Firstly Ti and Cu foils of each thickness were cut into rectangular strips with width of 1 mm and length of 10 mm by a sharp knife blade. The samples are then annealed at 300℃ for 5 hours in a vacuum furnace to
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eliminate residual stress. Without this process, the resultant experimental data would be too scattered to be analyzed due to the residual stress induced in the former step. Finally one end
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of each strip is glued to a rigid plate with 5 mm, while the other end extends out of the plate edge. Thus cantilever microbeams with width of 1 mm and length of 5 mm were made and subsequently mounted on the three-dimensional translation stage.
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The dynamic vibration measurement system consisting of a laser Doppler vibrometer
(LDV), a direct digital frequency synthesis (DDS) signal generator with minimum resolution of 0.01 Hz, a loudspeaker with power of 2 W, a computer, and a three-dimensional (3D) translation stage is illustrated in Fig. 1. The LDV probe with a laser beam diameter of 50 μm at the focused point is used for vibration measurement of the microbeam. And then the detected vibration response of the cantilever microbeam is displayed on the computer connected to the LDV. The DDS signal generator here can generate a continuous sinusoidal acoustic wave to excite the cantilever microbeam by the aid of loudspeaker. This kind of non-contact excitation excludes the influence of additional mass.
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Before the experiment, the distance and angular between the laser probe and the cantilever microbeam is elaborately adjusted to ensure the laser beam is focused on the free end of the microbeam. The cantilever microbeam is placed uprightly on the 3-D translation stage which can make the laser beam transmitted from the LDV be reflected the same way back to the laser probe. The vibration response of the cantilever microbeam is then detected and displayed on the computer connected to the LDV. In order to obtain the resonant frequency of the cantilever microbeam, the excitation frequency of the sinusoidal signal starts
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from a relatively low frequency less than the natural frequency and is increased gradually. Each time after increase, wait for 5 seconds to allow the response of the cantilever microbeam to stay stable, and then the amplitude of the vibration is recorded. When the excitation frequency approaches the resonant frequency, adjust the excitation frequency at the step of 0.1 Hz for sake of accuracy. The process continues until the recorded amplitude no longer increases, which indicates the excitation frequency has surpassed the resonant frequency.
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Thus a complete frequency-response curve near the resonant frequency is obtained. From the frequency-response curve, one can read the value of resonant frequency, and the error is within 0.1 Hz. Some typical frequency-response curves of Ti and Cu cantilever microbeams in different thicknesses are given in Fig. 4. It can be seen that the maximum magnitude of response amplitude is less than the corresponding thickness, also the frequency-response curve is symmetric near the peak of the curve. Conclusion can be made that vibration of the
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microbeams is in linear elastic range. Each specimen is measured at least three times and the experimental data are averaged, and for every single thickness, at least five specimens are
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prepared. Details of the experiment can be found in the previous work [55]. To reduce experimental error, the exact length and width of the cantilever microbeams are examined in an optical microscope to ensure the tolerance is within 0.05 mm before the
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tests. Furthermore, after the experiments, the actual thickness of Ti and Cu samples are measured through cutting a square hole vertically in the center of specimens with focused ion
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beam (FIB), as shown in Fig. 3(a) and (b). Besides, the surface roughness of each sample is tested in a laser scanning confocal microscopy (LSCM). A typical surface morphology is depicted in Fig. 3(c) and (d). The full data of thickness and roughness are listed in Table 1.
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One can see that the roughness of each sample is small enough to perform the experiment.
3 Length scale parameters fitting To obtain the length scale parameter, the equation of motion of size dependent microbeams are derived [57], i.e.
AW ( EI Al 2 )W (4) 0
(1)
Where l is the material length scale for modified couple stress theory, EI is the bending
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(2)
Substituting Eq. (2) into (1) and separating variables leads to
cos sn L cosh sn L +1=0
(n 1,2,3,...)
sn L 1.875， 4.694, 7.855, ... Then finally yields
EI Al 2 （sn L）2 , 4 AL
(n 1, 2, 3, )
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where
(3)
(4)
(5)
According to the classical Bernoulli-Euler beam theory [31], the natural frequency of a cantilever beam is given by
cn
EI 2 （s L） , AL4 n
(n 1, 2, 3, )
(6)
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Where subscript c in cn stands for classic solution.
Combining Eq. (5) and Eq.(6), the dimensionless frequency is obtained
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1 l 1 6( ) 2 c1 h
(7)
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Here, for simplicity, the Poisson’s effect is neglected [57], i.e. 0 .
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Effects of Poisson’s ratio and plain strain state For a slender beam with a large aspect ratio, the Poisson effect is secondary and may be
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neglected to facilitate the formulation of a simple beam theory [31], as demonstrated in the above section. For beams with relatively small aspect ratio, taking into account the influence of Poisson effect, Eq. (7) is rewritten as
1 6 l 2 1 ( ) c1 1+ h
(8)
Where substitution
is applied in Eq. (5).
E 2(1 )
(9)
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On the other hand, when the thickness of the thin beam is much smaller than its length and width, e.g. b/h>5, the displacement along the width direction is ignored hence plane strain state should be considered. In this case the bending rigidity EI in Eq. (5), (6) is transformed to EI , consequently the dimensionless frequency yields (1 2 )
(10)
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1 l 1 6(1- )( )2 c1 h
By fitting to the experimental data with Eq. (7) utilizing the least squares method, one can obtain the material length scale in MCST model, which gives l=1.442 μm for Cu and l=0.775 μm for Ti. In the case that Poisson’s effect is considered, the material length scale parameters are fitted via Eq. (8) as l=1.573 μm and l=0.877 μm for Cu and Ti respectively;
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And in the present case plain strain state is satisfied, from Eq. (10) one can obtain the material length scale parameter as l=1.661 μm for Cu and l=0.922 μm for Ti. In the last two situations the material length scale parameter is around 10% more than the corresponding case where Poisson’s ratio is ignored.
Also, the material length scale parameters for SGT can be solved based on the resultant experimental data if one assumes that all the three parameters are the same, i.e. l0 l1 l2 l
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. Of course it’s only for discussion, as pointed out by Lam [21]. The governing equation are determined as follow [55]:
4W 6W 2W K A 0 x 4 x 6 t 2
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where
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4 8 K I 2l02 l12 , S EI 2 Al02 Al12 Al22 5 15
(11)
(12)
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Taking advantage of the technique in literature [55], where differential quadrature
method (DQM) is applied to solve the governing equation and the method of minimization of
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least squares of the error between the experimental data and the SGT results is used to fit in the curve. The length scale parameters in SGT are obtained as 0.631 µm for Cu and 0.353 µm for Ti, respectively. Details of the calculation procedure are referred to literature [55]. It’s noted that there are higher-order boundary conditions for SGT however classic boundary conditions are employed in the above calculation for comparison to the result in literature [55]. Considering non-classic boundary conditions [54]and repeating the calculation procedure, the length scale parameters in SGT are as 0.604 µm for Cu and 0.311 µm for Ti. The errors are respectively around 10% and 5% comparing to the results obtained by classic boundary conditions. The dimensionless bending rigidity (the ratio of bending rigidity) is defined as
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EI Al 2 ( 1 )2 EI c1
(13)
The dimensionless frequency and the dimensionless bending rigidity with different thicknesses are shown in Fig. 5. As seen, the dimensionless frequency of Cu microbeams increases to as high as 1.6 times with decreasing the thickness of cantilever microbeam from 13.12 to 2.79 µm, while the dimensionless bending rigidity increases to about 2.6 times. Similarly, the dimensionless frequency and the dimensionless bending rigidity of Ti
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microbeams are increased to 1.38 times and 1.9 times when the microbeam thickness is decreased from 14.69 to 2.02µm. It indicates the inconsistency between the experimental natural frequencies and the classical solutions increases with the decreasing thickness, hence there exists size effects.
The fitted dimensionless frequencies curves based on the SGT, MCST and classical theory (l = 0) of Cu and Ti are displayed in Fig. 6. One can see that the SGT and the MCST
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give nearly the same curve, and that the experimental data is in good agreement with that predicted by higher order continuum theories.
The fitted elastic length scale parameters are shown in Table 2. For comparison, results obtained in the same condition as in the literature [55] are used. One can see that, compared to the material of epoxy, the material length scale of metallic materials are much smaller, about
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one-tenth [21, 31] and one-sixth [22] of each epoxy material. It reveals that the intrinsic structure between the metallic materials and the epoxy is different. The fitted curves of
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metallic (in the same condition) and non-metallic materials based on the MCST are depicted in Fig. 7. Since the SGT model gives nearly the same curve as the MCST, as seen in Fig. 6, the fitted curves based on SGT are omitted. One can see that the higher the length scale
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parameter is, the stronger the size effect becomes whether it is metallic or non-metallic materials. Moreover, there is an interesting observation from Table 2 that the value of length scale parameters for Cu and Ni are similar to each other when their densities are almost the
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same, although the Young’s modulus of Cu is merely half of that of Ni. However, Cu and Ti are similar in Young’s modulus but the density of Cu is twice of that of Ti. Dramatically, the
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length scale parameter in Cu is also nearly twice of Ti. This observation suits for length scale parameters based on both MCST and SGT. It seems mass density has greater influence on the length scale parameter than Young’s modulus. To further understand the relationship between length scale parameter and the material properties requires more experimental data in the near future.
4 Conclusion A standard and reliable method for determining the material length scale based on
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modified couple stress theory (MCST) is proposed in the paper. The method has been successfully applied to two metallic materials, Cu and Ti. As a result, evident elastic size effects are observed. By fitting to the experimental data, the length scale parameter of MCST is determined. Comparing the results of SGT and MCST, we find the value of length scale parameter in MCST is almost twice of that in SGT. The comparison between the epoxys and the metallic foils indicates a huge difference in the length scale parameters for these different materials.
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The reason could be related to the intrinsic material structures. Comparison between metals indicates that nickel foil has the strongest size effect, while Ti has the weakest size effect. And furthermore mass density seems to have greater influence on the length scale parameter than Young’s modulus.
Finally, compared to the static bending experiments, the method developed here is more convenient and economical to implement, thus can be used as a standard method for
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determining the material length scale in modified couple stress theory. The method will be useful for theoretical and numerical simulation of micro-structures and is of great importance for the design of MEMS/NEMS.
Acknowledgments
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This work was financially supported by the National Natural Science Foundation of
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China (Nos. 11772138, 11472114, 11572133, 11702103). DL also thanks the financial support of the Young Elite Scientist Sponsorship Program by CAST (No. 2016QNRC001).
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Fig. 1. Schematic diagram of vibration measurement system.
Fig. 2. Schematic diagram of the apparatus for electropolishing.
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Fig. 3. (a) and (b):Typical images of FIB thickness measurement of Cu and Ti;
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(c) and (d):Typical surface morphology of Cu and Ti(in laser color picture mode).
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Fig. 4. Amplitude-frequency curves of: (a) Ti, (b) Cu.
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Fig. 5. Dimensionless frequency for: (a) Ti, (b) Cu; Dimensionless bending rigidity for: (c) Ti, (d) Cu.
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Fig. 6. Mean values of the experimental data, SGT solution, MCST solution and the classical solution of the dimensionless natural frequencies for: (a) Ti, (b) Cu.
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Fig. 7. Size effect comparision of different materials, (a): Ni [55], Cu, Ti; (b): Epoxy [21], Epoxy [22] and SU-8 [22].
Table 1
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Ti
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Cu
Thickness/μm
Ra/nm
14.690
23.7
10.440
60.7
5.410
48.5
2.015
45.3
13.123
45.5
9.093
48.0
6.490
62.5
4.206
35.1
2.790
28.4
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Thickness and roughness of Ti and Cu foils.
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Table 2 Length scale parameters of each material.
Poisson's ratio
ρ/kg/m3
Ni[55]
207
--
8900
Cu
108
0.32
8900
Ti
108
0.31
4505
Epoxy[21]
1.44
--
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Epoxy[22]
3.93
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1.44
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--
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SU-8[22]
theory
l/μm
SGT MCST SGT MCST SGT
0.843 1.553 0.631 1.422 0.353
MCST SGT MCST SGT MCST SGT
0.775 11.01 17.60 4.35 7.75 1.39
MCST
2.50
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