Abstract
This paper presents two eigenfrequencybased damage diagnosis methods in a cantilever beam. The analytical relationship has been established between the eigenfrequency and damage parameters, including relative damage location and severity. On the premise that predamaged eigenfrequencies are known, a diagnosis algorithm without requirement of material properties is proposed based on change ratios of the first three eigenfrequencies. If predamaged eigenfrequencies are unfeasible to be acquired, a threecontour method based on only postdamaged eigenfrequencies is introduced to estimate damage parameters. The uniqueness of solution is discussed. Both the numerical simulation by the finite element method and the experiment on real beams are conducted and result in a good agreement between actual damage parameters and calculated values by using the proposed methods.
1. Introduction
Structural damages, caused by material aging, impact, fatigue, chemical attack and other unexpected mechanical loadings, may cause the performance degradation or even lead to the catastrophic failure in a structure. In the past few decades, a significant amount of analytical, numerical and experimental investigations has been carried out to detect damages at earliest possible stages through measuring and analyzing changes of modal properties on a damaged structure [14], based on the principle that a localized damage reduces the stiffness and increase the damping in the structure thus further decreases the eigenfrequency and alters the mode shape. Compared with nondestructive examination methods (NDE), such as Xray imaging, ultrasonic scans and eddy current testing, et al, the modalparameterbased damage diagnosis technique is more sufficient to cater for the needs of long range, quick global inspection and in service inspection of structures.
Eigenfrequency is one of the most popular modal properties used in damage identification due to its attractive characteristic of being relatively easy to be measured in a high precision. However, the frequencybased method can be only applied to typical structures and damages which can be theoretically modeled through mathematical approximations. Beam is one of the simplest and most commonly used structures, and a variety of complex structures are comprised of beams, hence the fundamental theory of the frequencybased damage identification technique is established on the beamtype structure, especially on the slender EularBernoulli beam. Usually, an open transverse edge crack is considered as the typical localized damage. The crack is theoretically equivalent to a massless linear torsional spring, and the beam is treated as two segmental beams connected by the spring. The earliest spring model was the axial spring proposed by Adams et al. [5], however, the theory to quantitatively analyze the equivalent spring stiffness did not be constructed. Then, a rotational elastic spring model was developed by Papadopoulos et al. [6], the equivalent spring stiffness of which was given as a function of the damage severity. Based on this model, Rizos et al. [7] arose an 8×8 determinantal equation relating the eigenfrequency, damage parameters (damage location and damage severity) and material properties (Young's modulus and density) for a cantilever beam. Other researchers then developed similar equations for beams with different boundary conditions. Ostachowicz et al. [8] deduced analytical expressions of equivalent spring stiffnesses of two different damages, open doubleside crack and open singleside crack. Their conclusions indicated that the equivalent spring stiffness is a function of only the damage severity, with no relationship with the damage location. They also constructed the mathematical relationship between the eigenfrequency, damage parameters and material properties in the form of a 12×12 determinantal equation for a cantilever beam with the assumption of two cracks. Liang et al. [9] proposed an approach based on any three eigenfrequencies to determine damage parameters in a cantilever beam or a simply supported beam. The method treats the crack as a rotational spring, plots the relationship curve between the equivalent stiffness and the damage location for each eigenfrequency, and determines the stiffness and the damage location in the intersection of three curves. The damage severity is then calculated from the relationship formula of the equivalent stiffness and the damage severity. This approach was then extended to stepped beams by Nandawana et al. [10] and geometrically segmented beams by Chaudhari et al. [11]. Owolabi et al. [12] presented a similar threecontour method based on any three eigenfrequencies. In this method, the threedimensional curved surface of each eigenfrequency in terms of the damage location and severity is obtained through the determinantal equation, and then the contour on the surface corresponding to the measured eigenfrequency value is projected onto the locationseverity coordinate plane. The intersection of three contours points out the damage location and severity.
These above mentioned methods detect the damage using postdamaged eigenfrequencies only, with no requirement of corresponding predamaged eigenfrequencies. However, in these methods, it is necessary to know material properties which are normally inconvenient to be measured in practice and solve a complicated determinantal equation. Plenty of effort has been devoted to frequencychangebased methods to overcome this problem. Cawlay et al. [13] proved that the ratio of eigenfrequency changes for two modes is only related with the damage location but no damage severity. Hearn et al. [14] also demonstrated that for a structure with a single damage, the variation ratio of squared eigenfrequency is a function of the damage location only. Narkis et al. [15] deduced a theoretical formula to describe relationship between the damage location and the ratio of eigenfrequency variations for a simply supported beam in the transverse vibration and the longitudinal vibration, and for a freefree beam in the longitudinal vibration. On the assumption that the crack is very small and leads to no volume change, Gudmundson [16] concluded a linear relationship between the fractional change of eigenfrequency and that of modal strain energy via the first order perturbation method. Based on Gudmundson's theory, Kim et al. [17, 18] studied the relationship between damage parameters and the variation ratio of eigenfrequency, and proposed an indicator for the single damage. The indicator is defined as the difference between the variation ratio of modal strain energy and that of eigenfrequency, and it reaches maximum at the damage position. Rubio [19] gave a relationship formula between the square ratio of the postdamaged eigenfrequency to the predamaged eigenfrequency and damage parameters for a simplysupported beam, and employed an optimization technique by minimizing a least square criterion to determine the damage location and severity. Sayyad et al. proposed an eigenfrequencychangebased damage diagnosis algorithm for beam structures with simply supported [20] and cantilever [21, 22] boundary conditions. They got a right algorithm for the simply supported beam, but mistook the mode shape of the simply supported beam for that of the cantilever beam and thus drew an erroneous conclusion for the cantilever beam. In frequencychange based methods, no material property is demanded, but an accurate knowledge of predamaged eigenfrequencies is indispensable. Unfortunately, it is unfeasible to meet the healthy structure in many real applications. Normally, the finite element method (FEM) is employed to model the structure, but as we know, the numerical simulation asks for material properties.
In this paper, two methods are proposed to diagnose the damage in a cantilever beam, for situations with and without eigenfrequencies of the indefective beam. The relationship formula between damage parameters and the variation ratio of eigenfrequencies is established, with no material property. If eigenfrequencies of the intact beam can be acquired, an eigenfrequencychangebased method is presented. Algebraic equations based on the first three eigenfrequencies, related to the damage location only, are deduced from the relationship formula. The damage location is indicated by solving the equations, and then fed to the relationship formula to determine the damage severity. Analysis of the uniqueness of solution to the algebraic equations is carried out, which is always lacking in other eigenfrequencybased approaches. If frequencies of the intact beam are absent, a postdamagedeigenfrequencybased threecontour method is introduced to determine the damage location and severity. This method gains an advantage of being materialpropertyfree over other threecurve methods. Numerical simulations and experiments are conducted to verify the proposed methods. The results show that an accurate evaluation of both the damage location and the damage severity can be achieved with an error which is acceptable in practical applications by using the proposed methods.
2. Theoretical analysis for damage identification
2.1. Eigenfrequencychangebased method
A typical EulerBernoulli cantilever beam damaged by a small discrete singleside crack is considered, and a Cartesian coordinate system $x$$y$$z$ is established, as shown in Fig. 1. The length, height and width of the beam are $L$, $H$ and $B$, respectively. The crack of a depth $R$, which is supposed to remain open during vibrations, is at the position of distance $D$ to the clamping end. Normally, the relative position $e=D/L$ is used to represent the dimensionless crack location and the relative depth $\alpha =R/H$ is used to represent the dimensionless crack severity. Young’s modulus and mass density are represented by $E$ and $\rho $, respectively. For the convenience of deduction and calculation, another Cartesian coordinate system $u$$v$$w$ is established as well, in which $u=x/L$, $v=y/L$, $w=z/L$, hence the length, height and width of the beam in the new coordinate system are $l=$ 1, $h=H/L$, $b=B/L$, respectively.
Fig. 1An EulerBernoulli cantilever beam model with a single edge crack
Presuming the crack is small enough to cause no volume change, the following relationship between the fractional change of eigenfrequency $f$ and that of modal strain energy is proposed by Gudmundson [16]:
where ${f}_{i}$ and ${f}_{ci}$ are the angular eigenfrequencies before and after the crack occurrence, and subscript $i$ demotes the $i$th^{}mode, respectively. Considering that the eigenfrequency difference is very small, the sum of ${f}_{i}$ and ${f}_{ci}$ is feasible to be regarded as a double value of ${f}_{i}$, hence the following relationship formula can be acquired:
where $\u2206{f}_{i}={f}_{i}\uff0d{f}_{ci}$ is the eigenfrequency change. In order to satisfy the assumption, usually, the relative depth $\alpha $ is restricted in the region (0, 0.5). The restriction is reasonable for damage detection at earliest possible stages.
The $n$th modal strain energy of an undisturbed EulerBernoulli beam in the $u$$v$$w$ coordinates is given as follows [17]:
where ${\phi}_{i}$ is the $i$th mode shape.
For an EulerBernoulli beam structure with an edge crack under bending, the stress intensity factor ${K}_{I}$ is [23]:
where ${F}_{I}\left(\alpha \right)$ is a geometrical factor expressed as Eq. (5) and $\sigma $ is the stress level given by Eq. (6) with regard to a plane state of stress:
Hence, the strain energy density function ${J}_{s}$ can be obtained as follows:
The decrease of modal strain energy is:
where $A$ is the area of the crack damage. Since $dA=bdr$:
Since $\alpha =r/h$, it’s convenient to convert Eq. (9) to the following form:
Substituting Eq. (3) and (10) into Eq. (2), the following equation can be got:
where ${G}_{i}={\int}_{0}^{1}{\left[{\phi}_{i}^{\text{'}\text{'}}\left(u\right)\right]}^{2}du$ and ${g}_{i}={\left.{\left[{\phi}_{i}^{\text{'}\text{'}}\left(u\right)\right]}^{2}\right}_{u=e}$.
The mode shape of an EulerBernoulli cantilever beam is given as:
where:
Therefore:
For the first mode ($i=$ 1), ${\beta}_{1}=$ 1.875, thus ${G}_{1}={\int}_{0}^{1}{\left[{\phi}_{i}^{\text{'}\text{'}}\left(u\right)\right]}^{2}du=$ 12.3623. Similarly, ${G}_{2}=$ 485.52, ${G}_{3}=$ 3806.2. Since ${\alpha}^{2}f\left(\alpha \right)$ is a constant for a certain crack, it can be eliminated in division between any pair of eigenfrequency change ratios. Therefore, the following equations can be obtained:
The relative damage location $e$ can be determined by solving the equations using the NewtonRaphson method.
2.2. Analysis of uniqueness of solution
Each subequation in the above mentioned Eq. (14) can be solved independently, but each subequation has several solutions. Let ${g}_{a}\left(u\right)={g}_{1}\left(u\right)/{g}_{2}\left(u\right)$, ${g}_{b}\left(u\right)={g}_{2}\left(u\right)/{g}_{3}\left(u\right)$ and ${g}_{c}\left(u\right)={g}_{1}\left(u\right)/{g}_{3}\left(u\right)$, then the logarithmic values of ${g}_{a}\left(u\right)$, ${g}_{b}\left(u\right)$ and ${g}_{c}\left(u\right)$ in the $u$interval (0, 1) are plotted in Fig. 2.
The function ${g}_{a}\left(u\right)$ is monotone increasing in the interval (0, 0.217] and monotone decreasing in the interval (0.217, 1), as shown in Fig. 2(a). The maximum value of ${g}_{a}\left(u\right)$ in the interval (0.364, 1) is smaller than that in the interval (0, 0.217], thus Eq. (14a) has only one solution in the interval (0.364, 1). If the damage is located in this interval, there is only one simultaneous solution of three subequations. In the interval (0, 0.364], Eq. (14a) has two solutions, one of which in the interval (0, 0.217] and the other is in (0.217, 0.364].
As shown in Fig. 2(b), values of ${g}_{b}\left(u\right)$ in (0, 0.167] and (0.167, 0.364] are different, considering that Eq. (14a) has only one solution in the interval (0, 0.167], thus if the damage is located in (0, 0.167], it can be determined by the simultaneous solution of Eqs. (14a) and (14b). In the interval (0.167, 0.364], Eq. (14b) also has two solution, in the intervals (0.167, 0.217] and (0.217, 0.364], respectively.
In that case, consider values of ${g}_{c}\left(u\right)$ in the interval (0.167, 0.364]. As shown in Fig. 2(c), ${g}_{c}\left(u\right)$ is monotone decreasing in the interval (0.167, 0.318], and the minimum of ${g}_{c}\left(u\right)$ in (0.167, 0.318] is larger than that in (0.318, 0.364]. That means there exist only one simultaneous solution of the three subequations in (0.167, 0.318]. In (0.318, 0.364], ${g}_{c}\left(u\right)$ decreases first and then increases, therefore Eq. (14c) may have two solutions. However, either of Eq. (14a) and (14b) has only one solution in this interval, thus the three subequations also have only one simultaneous solution.
To sum up, there exist one and only one solution for Eq. (14) in the $u$interval (0, 1), indicating the relative damage location $e$.
According to Eq. (11), the variation of normalized postdamaged eigenfrequency versus relative damage location can be obtained, as shown in Fig. 3. The normalized postdamaged eigenfrequency is defined as the ratio of postdamaged to predamaged eigenfrequency. It’s worth noting that the exact value is determined not only the relative location but also other parameters in Eq. (11), thus the curve in Fig. 3 shows only the general variation trend versus relative location. Evidently, all the three normalized eigenfrequencies approach to 1 near the free end. The second normalized eigenfrequency approaches to 1 while $e$ is close to 0.217 and the third normalized eigenfrequency approaches to 1 while $e$ is close to 0.133 or 0.5. Being equal to 1 means that the predamaged and postdamaged eigenfrequencies are the same. In practical applications, when the damage is located near $e=$ 1, 0.133, 0.217 or 0.5, the measured postdamaged eigenfrequency may be higher than the predamaged eigenfrequency because of the measurement error, thus generally speaking, a tiny plus value of $\u2206{f}_{i}/{f}_{i}$ ought to be chosen for calculation instead of the measured value.
Fig. 2Variation curves of gau, gbu, gcu versus relative damage location
a)${g}_{a}\left(u\right)$
b)${g}_{b}\left(u\right)$
c)${g}_{c}\left(u\right)$
The damage severity $\alpha $ can be computed by Eq. (11) after the acquirement of $e.$ Any eigenfrequency decreases with the increase of $\alpha $ due to the monotone increasing function ${\alpha}^{2}f\left(\alpha \right)$ in (0, 0.5), as shown in Fig. 4, thus $\alpha $ is able to be determined via any subequation.
2.3. Eigenfrequencybased threecontour method
If the predamaged eigenfrequency is absent, transform Eq. (11) into:
hence, the postdamaged eigenfrequency ${f}_{ci}$ is:
Fig. 3General variation trend curve of first, second and third normalized postdamaged eigenfrequency versus relative damage location
a) First
b) Second
c) Third
Fig. 4General variation trend curve of normalized postdamaged eigenfrequency versus relative damage depth
As we know, the analytic eigenfrequency of an undisturbed EulerBernoulli beam is:
therefore, Eq. (15) is equivalent to the following:
In that case, it can be obtained:
Since Eq. (19) is deduced from Eq. (11), there is only one common solution in the $u$interval (0, 1) and $\alpha $interval (0, 0.5) for the three subequations, notwithstanding that each subequation has infinitely many solutions. By using the threecontour method, damage parameters can be determined with the first three eigenfrequencies and dimensional sizes of the beam. Material properties are needless, which are more difficult to be measured than dimensional sizes in practice.
3. Numerical simulation
The proposed two methods, one based on the eigenfrequency change and one based on only the postdamaged eigenfrequency, were testified by both the numerical simulation and the experiment. An intact steel cantilever beam and a series of damaged beams, of length $L=$ 330 mm, width $B=$ 15 mm and height $H=$ 12 mm, were modeled by FEM. Young’s modulus $E=$ 7803 kg/m^{2}, mass density $\rho =$ 207 GPa and Poisson's ratio $\nu =$ 0.3, respectively. Each damaged beam contained an open singleside notch crack, but on different beam the crack was of different depth at different position. The first three eigenfrequencies of the undamaged beam are 81.466 Hz, 507.77 Hz and 1409.6 Hz, respectively, and that of cracked beams are listed in Table 1.
Table 1First three eigenfrequencies of damaged beams simulated by FEM
$e=$0.2  $e=$0.3  e=0.4  $e=$0.5  $e=$0.6  $e=$0.7  $e=$0.8  
$\alpha =$0.1  
1st eigenfrequency  81.132  81.244  81.331  81.393  81.434  81.457  81.468 
2nd eigenfrequency  507.75  507.39  506.43  505.80  505.95  506.68  507.41 
3rd eigenfrequency  1408.0  1405.1  1407.3  1409.6  1406.6  1403.5  1405.7 
$\alpha =$0.2  
1st eigenfrequency  80.223  80.635  80.957  81.191  81.342  81.427  81.466 
2nd eigenfrequency  507.71  506.32  502.75  500.42  500.97  503.67  506.41 
3rd eigenfrequency  1403.4  1392.8  1401.0  1409.6  1398.6  1387.0  1394.7 
$\alpha =$0.3  
1st eigenfrequency  78.67  79.578  80.301  80.83  81.177  81.371  81.458 
2nd eigenfrequency  507.62  504.47  496.47  491.29  492.4  498.37  504.62 
3rd eigenfrequency  1395.6  1372.4  1390.5  1409.5  1385.3  1359.7  1375.4 
$\alpha =$0.4  
1st eigenfrequency  76.257  77.894  79.232  80.233  80.899  81.275  81.441 
2nd eigenfrequency  507.48  501.58  486.79  477.31  478.98  489.72  501.60 
3rd eigenfrequency  1383.5  1342.1  1374.8  1409.4  1365.7  1319.2  1344.0 
In the eigenfrequencychangebased approach, the crack location was calculated using Eq. (14), and then it is put into Eq. (11) to determine the crack depth. Results are listed in Table 2. Relative errors are calculated and listed as well, which are very small and demonstrate the accuracy of the proposed method.
Table 2Damage parameters calculated by the eigenfrequencychangebased method
$e=$0.2  $e=$0.3  e=0.4  $e=$0.5  $e=$0.6  $e=$0.7  $e=$0.8  
$\alpha =$0.1  
Calculated $e$  0.200  0.300  0.402  0.498  0.600  0.697  0.806 
Relative error  0  0  0.5 %  0.4 %  0  0.4 %  0.75 % 
Calculated $\alpha $  0.100  0.100  0.099  0.097  0.102  0.102  0.098 
Relative error  0  0  1 %  3 %  2 %  2 %  2 % 
$\alpha =$0.2  
Calculated $e$  0.199  0.299  0.402  0.496  0.601  0.696  0.796 
Relative error  0.5 %  0.33 %  0.5 %  0.8 %  0.17 %  0.57 %  0.5 % 
Calculated $\alpha $  0.203  0.203  0.199  0.201  0.201  0.199  0.198 
Relative error  1.5 %  1.5 %  0.5 %  0.5 %  0.5 %  0.5 %  1 % 
$\alpha =$0.3  
Calculated $e$  0.200  0.299  0.403  0.493  0.598  0.701  0.794 
Relative error  0  0.33 %  0.75 %  1.4 %  0.33 %  0.14 %  0.75 % 
Calculated $\alpha $  0.299  0.300  0.300  0.297  0.297  0.305  0.293 
Relative error  0.33 %  0  0  1 %  1 %  1.7 %  2.3 % 
$\alpha =$0.4  
Calculated $e$  0.200  0.299  0.403  0.490  0.596  0.705  0.790 
Relative error  0  0.13 %  0.75 %  2 %  0.67 %  0.71 %  1.25 % 
Calculated $\alpha $  0.392  0.396  0.403  0.392  0.394  0.408  0.395 
Relative error  2 %  1 %  0.75 %  2 %  1.15 %  2 %  1.25 % 
The threecontour method was validated using eigenfrequencies of the defective beam with a relative crack location $e=$ 0.6 and a relative crack depth $\alpha =$ 0.3 provided above, as an instance. Fig. 5 shows three curved surfaces calculated via the three subequations in Eq. (19). The $x$ and yaxes represent relative crack location and depth, respectively. The $z$axis in Fig. 5(a), (b) and (c) represent ${f}_{c2}/{f}_{c1}$, ${f}_{c3}/{f}_{c2}$ and ${f}_{c3}/{f}_{c1}$, respectively. For the conditions $e=$ 0.6 and $\alpha =$ 0.3, ${f}_{c2}/{f}_{c1}=$ 6.07, ${f}_{c3}/{f}_{c2}=$ 2.81 and ${f}_{c3}/{f}_{c1}=$ 17.07. The corresponding contours are projected onto the $e$$\alpha $ plane and shown in Fig. 6, as the solid line, dotted line and dash dotted line, respectively. The contours intersect in the point $e=$ 0.624 and $\alpha =$ 0.318. The relative errors are 4 % and 6 %, respectively, which denote a high precision of the proposed method.
Actually, the three contours are generally impossible meet in an exact point. It’s more frequent that each pair of contours intersect and finally three intersection points generate a triangle. Therefore, commonly, the triangular geometric center is selected as the intersection point of the three contours.
4. Experimental Illustration
Experiments were conducted to demonstrate the proposed algorithms. An uncracked and two cracked aluminum cantilever beams were manufactured with the same dimensions of 270 mm×10 mm×10 mm and material properties. An artificial single notch crack was made on each defective beam by wirecutting technique with the relative depth $\alpha =$ 0.4. The crack on one beam is at position of 108 mm to the clamping end and the other is 189 mm, which means that the crack location e are respective 0.4 and 0.7.
The shock test was implemented to determine the first three eigenfrequencies of each cantilever beam. A pendulum applied a shock excitation to the beam at the position of 10 mm from the free end, and a selfsynchronizing multipoint laser Doppler vibrometer [24] was employed to measure the outofplane surface displacement variation of four sampling points on the beam, as shown in Fig. 7. Laser Doppler vibrometer is a noncontact measurement tool, applying no accessional mass and influence to the specimen. When the pendulum almost hit the beam, the vibrometer was triggered to record the outofplane displacement, from the spectrum of which the first three eigenfrequencies of the corresponding beam can be extracted [24]. In order to eliminate error, an average value of eigenfrequencies obtained from the four sampling points was applied to the crack parameter calculation. All the measured eigenfrequencies and calculated crack parameters are shown in Table 3. Relative errors are computed and tabled, as well.
Fig. 5Variation curved surface of fc2/fc1, fc3/fc2, fc3/fc1 versus relative damage location and depth
a)${f}_{c2}/{f}_{c1}$
b)${f}_{c3}/{f}_{c2}$
c)${f}_{c3}/{f}_{c1}$
Fig. 6Contours and their intersection under the damage conditions e= 0.6 and α= 0.3
Errors in the experiment are relatively larger than that in the numerical simulation. It is generally believed that the main error source is the difference between the real structure and its theoretical mathematic model based on the theory of EulerBernoulli beam and the rotational spring approximation of the crack damage. The difference brings forth an inherent error in the algorithm. The FEM model itself is established on the theory of EulerBernoulli beam, hence in numerical simulation this error is reduced. On the other hand, although a noncontact optical metrology method is adopted, the experimental setup induced measurement error is still inevitable. However, the relative errors in the experiments are less than 20 %, which can be accepted in engineering applications.
Fig. 7Experimental layout for the measurement of eigenfrequencies of a cantilever beam by LDV
Table 3Eigenfrequencies of test beams and damage parameters calculated by the proposed two methods
Intact  $e=$0.4  $e=$0.7  
1st eigenfrequency  105.6  103.3  105.4 
2nd eigenfrequency  663.6  637.7  646.8 
3rd eigenfrequency  1865.7  1813.2  1735.9 
$e$ calculated by Eq. (14)  –  0.41  0.68 
Relative error  –  2.5 %  2.85 % 
$\alpha $ calculated by Eq. (11)  –  0.37  0.33 
Relative error  –  7.5%  17.5 % 
$e$ calculated by Eq. (19)  –  0.41  0.74 
Relative error  –  2.5 %  5.7 % 
$\alpha $ calculated by Eq. (19)  –  0.34  0.37 
Relative error  –  15 %  7.5 % 
5. Conclusions
The location and the severity of a structural damage can be assessed by the change of eigenfrequency before and after the damage occurrence. Cantilever beam is one of the mostly common used structure types in engineering fields. In this paper, two eigenfrequencybased methods are developed to identify the damage location and severity of a cantilever beam. One is based on the eigenfrequency variation before and after the damage appearance, and the other is a threecontour method based on eigenfrequencies of the defective beam only. Both the two methods enjoy the virtue of demanding no knowledge of material characteristics of the structure. The threecontour method is less precise but more universal than the eigenfrequencyvariationbased method, since it needs no frequency of the healthy structure. The uniqueness of solution is analyzed and confirmed, and numerical simulative and experimental verifications are carried out to show that the proposed methods are simple, effective and accurate.
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The authors would like to acknowledge the financial support provided by the National Science Foundation of China (NSFC) under Grant No. 11502257, the key subject “Computational Solid Mechanics” of CAEP, and the Science and Technology Development Foundation of CAEP under Grant No. 2014B0101009.