J Fail. Anal. and Preven. https://doi.org/10.1007/s11668-018-0418-4 TECHNICAL ARTICLE—PEER-REVIEWED Failure Analysis of Composite Mono Leaf Spring Using Modal Flexibility and Curvature Method N. I. Jamadar . S. B. Kivade . Rakesh Raushan Submitted: 25 October 2017 / in revised form: 2 January 2018 Ó ASM International 2018 Abstract Post-failure analysis of composite structures by damage identification at pre-stage has gained considerable interest in recent years. In the paper, modal flexibility and modal curvature methods have been used for premature failure analysis of composite mono leaf spring through analytical and finite element approaches. Initially, finite element model of healthy and cracked spring is used to evaluate the eigenvalue and eigenvectors. Subsequently, change in flexibility and absolute curvature difference among both springs are evaluated after mass normalization. The presence, location and severity of crack in a spring are identified through differences in local flexibilities and change in absolute modal curvatures. The modal curvature method predicts location and severity of crack more precisely than the modal flexibility method. M*e Ke K*e le K K* M M* U [F]hx [F]dx Uxi Keywords Failure analysis  Modal flexibility  Modal curvature  Composite mono leaf spring  Analytical  Healthy and cracked spring List of symbols Me Elemental mass of healthy beam N. I. Jamadar (&) Research Resource Center, VTU, Kalaburagi Region, Karnataka, India e-mail: jamadar94@gmail.com S. B. Kivade Basavakalyan Engineering College, Basavakalyan, Karnataka, India e-mail: sbkivade@gmail.com R. Raushan Dr. D. Y. Patil Institute of Technology, Pimpri, Pune, Maharashtra, India e-mail: prabhatpankaj1989@gmail.com D[F] M(x) M*(x) K(x) K*(x) D[K] Uh1, Uh2, Uh3 and Uh4 Ud1, Ud2, Ud3 and Ud4 Kh1, Kh2, Kh3 and Kh4 Elemental mass of cracked beam Elemental stiffness of healthy beam Elemental stiffness of cracked beam Elemental length of beam Assembled stiffness matrix of healthy beam Assembled stiffness matrix of cracked beam Assembled mass matrix of healthy beam Assembled mass matrix of cracked beam Eigenvector Flexibility matrix of healthy beam at distance x Flexibility matrix of cracked beam at distance x Mass normalized modal vector at distance x Change in flexibility Bending moment of healthy beam at distance x Bending moment of cracked beam at distance x Curvature of healthy beam at distance x Curvature of cracked beam at distance x Absolute modal curvature difference Are the normalized eigenvectors of healthy beam for 1st, 2nd, 3rd and 4th modes, respectively Are the normalized eigenvectors of cracked beam for 1st, 2nd, 3rd and 4th modes, respectively Are the modal curvatures of healthy beam for 1st, 2nd, 3rd and 4th modes, respectively 123 J Fail. Anal. and Preven. Kd1, Kd2, Kd3 and Kd4 Are the modal curvatures of cracked beam for 1st, 2nd, 3rd and 4th modes, respectively Introduction Study of composite structural fatigue failures in service condition has gained significance in scientific research community. The failure of structures occurs due to reduction in local stiffness and leads in lowering the load carrying capacity. The damage mechanism in these structures is complex in nature than the conventional materials. The defects developed during manufacturing or in service condition leads to low fatigue cycle failures in composite structures. These defects include micro-cracks, delamination such as inter-laminar or de-bonding, micro-voids and inclusions. These defects grow in progression during fatigue loading. The progressive assessment of such damages at the intermediate status is one of the challenging tasks. He et al. [1] explored damage identification in composite beams through curvature mode difference via central difference operation with single and multiple damages on it. The experimental and simulation results showed the changes in modal difference between intact and damaged beams. The presence, location, size of damage identified accurately. Zhou et al. [2] proposed numerical method to locate the damage in ocean drilling riser by modal curvature difference which showed accuracy in locating the damages of intact and damaged element. Yang et al. [3] presented Fourier spectral-based method, an alternate to modal curvature method used for locating the damage of any structures. The comparison of both the methods has been demonstrated by experimental and numerical analysis for accuracy. However, found weak in detecting small damages. Yang et al. [4] proposed discrete Fourier transform-based modal curvature and scale wavenumber domain filtering method to inspect the damage in carbon-reinforced polymer beam with and without crack. Researcher concludes the traditional modal curvature method weaker in dealing with the noisy data which deviates the accuracy in detection of damage. Analytical method carried out and validated with the experimental work. Dessi and Camerlengo [5] overviewed various damage identification methods based on the natural frequency, curvature mode shape and modal strain energy. Sensitivity and performance analysis has been carried out to adjacent points of the damage. Dawari and Vesmawala [6] applied both modal curvature and modal flexibility methods to identify damage in reinforced concrete beam and showed that modal curvature changes are high at region near the damage. Russo [7] also investigated the homogeneity of the glass fiber- 123 reinforced polymer structural beam elements with different cross sections. The damage location and its extent in these structures were found based on their dynamic characteristics by curvature mode shapes. Ekinovic et al. [8] determined the damage present in the cantilever beam through modal parameters. The numerical model of beam has been formulated to assess its dynamic modal characteristics. The modal curvatures and displacement mode shapes have been calculated by central difference approximation. The results showed that the changes in mode shape characteristics are the good indicators for location and severity of damage. Lestari and Qiao [9] investigated experimentally the damage location in fiber-reinforced polymer (E-glass) sandwich beam based on curvature mode shape with piezoelectric sensor and showed the lower modal curvature decides accurately the damage location than higher modal curvatures. Zhua and Xub [10] presented numerical and experimental approach on 10-mono coupled periodic structure by sensitivity-based method for damage localization and its severity. To locate and quantify the damage, curvature mode shape/slopes, natural frequencies are used, respectively. It has been noticed that the modal curvatures are more sensitive and accurate in locating the damage than modal slopes. Lee and Eun [11] derived analytical methods to update the flexibility matrix by the expanded full degree of freedom. The detection of damage carried out based on updated flexibility matrix. It is concluded that the second derivative of flexibility is accurate in yielding results as compared to traditional method. Sung et al. [12] presented new approach of damage detection over cantilever beam using damage induced inter-storey deflection estimated by modal flexibility. Gaith [13] determined the effect of crack location, depth, fiber orientation and volume fraction on flexibility of graphite/epoxy composite beam and determined the effect of these parameters on natural frequency and mode shapes. Reynders and Roeck [14] determined the local stiffness directly through flexibility to identify the presence and location of damage in a reinforced concrete beam. Yu and Yin [15] proposed a method for damage detection over a portal frame by finite element model updating. This methodology aimed to minimize the discrepancies among the experimental and analytical modal parameters. Abdo and Hori [16] studied numerically the cantilever steel plate for damage detection using changes in the rotation of mode shapes instead of modal flexibility and curvature mode shapes. It has been proved that the change in rotation mode shape is the sensitive indicator for single and multiple damages. Pandey and Biswas [17] used modal flexibility method for locating damage in wide flanged beams and concluded that only lower-frequency modes are suitable in locating the damage in structures. Yuen [18] presented a study of damage size and its location by changes in J Fail. Anal. and Preven. eigenvalue and eigenvector in damaged cantilever beam. The changes in the eigenvalue and eigenvectors have showed the indication of location and extent of damage. Most of the research articles investigated the damage presence, location and severity in a structure based on modal curvatures, flexibility, slopes and rotations. The main focus of these researchers was on steel plates, frames, concrete beams and fiber-reinforced structures. The extensive study on composites at early stage of damage identification on automotive structural parts such as suspension leaf springs, chassis and body frames is essential. Details of Composite Mono Leaf Spring In the present work, composite mono leaf spring of utility vehicle is analyzed. It is single leaf spring with T-3 tapered trapezoidal cantilever [19] which has maximum width 125 mm, minimum width 72 mm, maximum thickness 18 mm and minimum thickness 12 mm. It is made of Eglass/epoxy material with unidirectional laminated fibers of high strength/weight ratio than the conventional leaf spring. Table 1 shows mechanical properties of unidirectional E-glass/epoxy composite material, and Fig. 1 shows the actual mono composite leaf spring. Artificial Crack Location In practice, the failures of leaf springs are due to low fatigue cycles resulted in weakening the portions of it leading to premature failure. To understand the presence, location and severity of damage, an artificial crack is created on the healthy spring. Table 1 Mechanical properties of unidirectional E-glass/epoxy material Sr. no. Properties Value 1. Density (Kg/m3) 2000 2. Young’s modulus X direction (MPa) 50,000 3. Young’s modulus Y direction (MPa) 10,000 4. Young’s modulus Z direction (MPa) 10,000 5. Poisson’s ratio XY 0.3 6. Poisson’s ratio YZ 0.4 7. Poisson’s ratio XZ 0.3 8. Shear modulus XY (MPa) 5000 9. Shear modulus YZ (MPa) 3846.2 10. Shear modulus XZ (MPa) 5000 11. Ultimate tensile strength (MPa) 1390 Initially, finite element analysis of healthy composite leaf spring is conducted to identify the damage sensitive region to create artificial crack on it. This region is located 130 mm from the center and 380 mm from the eye end. The crack dimensions are selected using the relation [20]. It is a = a/w where a = crack ratio (0.2–0.8), a = crack length, w = width in direction of crack propagation. As the composite material is orthotropic behavior in nature, it is planned to check the crack behavior by creating small crack of size 2 mm length and 2 mm depth on the surface. The crack length is further increased to observe the effect of crack size on propagation. The cracked spring is subjected to set number of fatigue cycles of 500, 1000 and so on in succession until the crack evolution. After every 500 fatigue cycles, static testing has been carried out to record the strains and corresponding stiffness for maximum load of 5000 N. Finally, the full length crack along the width has been created. Figure 2 illustrates the same. Theoretical Background The modal parameters such as natural frequencies, mode shapes, flexibility, curvatures and modal rotations are used for damage detection. Among these parameters, modal flexibility and curvature mode shape methods are powerful techniques for inspection of damage. The differences in numerical values in the parameters are compared with healthy spring. Cantilever Beam Finite Element Model The trapezoidal mono composite leaf springs are modeled as healthy and cracked cantilever beams by finite element method as shown in Figs. 1 and 2. It has been divided into four equal parts by considering equivalent width and thickness (t) across length of beam. The thickness has been reduced to account for reduction in the cracked area over the cracked beam. The numbers mentioned for each portions are the modal displacements, i.e., 1, 3, 5, 7, 9 and 2, 4, 6, 8, 10 are modal rotations, respectively. The numerical analysis is carried out by considering modal displacements only (Figs. 3 and 4). The equation of motion for ‘n’ degrees of freedom of a structure can be written as   ::    :  M x þ C x þ K ¼ 0: The healthy and cracked springs are modeled by considering consistent mass. The elemental masses for healthy and cracked beams can be expressed as 123 J Fail. Anal. and Preven. Fig. 1 Composite mono leaf spring Fig. 2 Artificial crack on the spring Artificial Crack 118.75 Fig. 4 Finite element model of cracked spring 118.75 78.63 118.75 105.125 t= 14.25 9,10 7,8 t=15.625 5,6 3,4 t=12.75 123 t= 14.25 118.75 118.38 1,2 91.88 105.125 118.38 t=12.75 118.75 5,6 3,4 1,2 118.75 t=17.25 118.75 78.63 118.75 91.88 Fig. 3 Finite element model of healthy spring 9,10 7,8 t=13.625 t=17.25 J Fail. Anal. and Preven. 2 3 156 22le 54 13le qAle 6 4l2e 13le 3l2e 7 6 22le 7 Me ¼ 4 54 13le 156 22le 5 420 2 13l 4l2 2 e 3le 22le 3 156 22le 54 13le qAle 6 4l2e 13le 3l2e 7 6 22le 7: Me ¼ 4 54 13le 156 22le 5 420 13le 3l2e 22le 4l2 The elemental stiffness for both the beam can be expressed as 2 3 12 6le 12 6le EI 6 6le 4l2e 6le 2l2e 7 7 Ke ¼ 3 6 4 12 6le 5 le 12 6le 6le 2l2e 6le 4l2 2 3 12 6le 12 6le EI 6 6le 4l2e 6le 2l2e 7 7: Ke ¼ 3 6 4 12 6le 5 le 12 6le 6le 2l2e 6le 4l2 The generalized characteristic equation or eigenvalue problem for ‘n’ degrees of freedom of healthy and cracked beam can be written as   ½ðK Þ  ki ðM Þ ¼ 0 and ½ðK Þ  ki ðM Þ ¼ 0 Normalization of mass matrix of healthy and cracked beam is given as UTMU and UTM*U. Modal Curvature Method It is also one of the damage evaluation methods to detect the presence, location and severity in a damaged structure or component by comparison with the healthy structures. The modal curvatures for healthy and damaged structures are determined using second-order central difference method in modal displacement field. There is direct concurrency of damage location with the modal curvature fields found, and the magnitude is directly proportional to severity of damage. This method is effective for the lower modal frequencies. For instance, consider a cantilever beam with EIas flexural rigidity, subjected to bending moment as M(x) and curvature as K(x). Here, ‘x’ indicates the distance of any point from fixed end of the beam. The curvature for healthy cantilever beam can be given as KðxÞ ¼ And the curvature at a distance x for cracked cantilever beam can be given as K  ðxÞ ¼ KðxÞ ¼ ½F hx ¼ 1 X 1 U UT 2 xi xi x i i¼1 ½F dx ¼ 1 X 1 U UT : 2 xi xi x i i¼1 The change in flexibility matrix among the beams will be D½F  ¼ " ½F hx ½F dx # " # 1 1 X X 1 1 ¼ Uxi UxiT  Uxi UxiT : x2i x2i i¼1 i¼1 hx M  ðxÞ EI The mode shape curvatures at local level for healthy and cracked beam are computed as follows based on the central difference approximation. Modal Flexibility Method This method is one of the modal techniques to identify the presence, location and severity of damage in structures. It includes the determination of natural frequencies, mass normalized mode shapes and flexibility matrix of structures. It yields better results for lower modes and converges with the higher modes. The flexibility matrix of healthy and cracked beam at distance x along its length can be calculated as MðxÞ EI /ðj þ 1Þi  2/ji þ /ðj  1Þi l2 K  ðxÞ ¼ /ðj þ 1Þi  2/ji þ /ðj  1Þi l2 where i mode shape number, j node number, uji modal displacement of node j at mode i and l distance between nodes. The absolute modal curvature difference between the healthy and cracked beam approximates the location of damage.   uðj þ 1Þi  2uji þ uðj  1Þi D½K  ¼ ½K  ð xÞ  K ð xÞ ¼ l2 dx   uðj þ 1Þi  2uji þ uðj  1Þi  l2 hx Based on the higher value of curvature difference, the location of the crack can be identified in a beam. Numerical Method dx Here, xi is the ith natural frequency. The increase in the flexibility of corresponding element is used to detect the presence and the location of damage. To detect of presence, location and severity of crack in a cracked spring, the modal flexibility and modal curvature methods are applied. 123 J Fail. Anal. and Preven. 2 Modal Flexibility Method The step-by-step procedure to calculate modal flexibility of both the springs is given below. Calculation of Assembled Mass and Stiffness Matrix The elemental stiffness and mass matrices are assembled for healthy springs as given below. 2 3 16408:68 9082:8 0 0 6 9082:8 7 19547:52 10464:72 0 7 K¼6 4 0 10464:72 22515 12050:28 5 0 12050:28 12050:28 2 0 3 0:265 0:046 0 0 6 0:046 0:259 0:044 0 7 7 M¼6 4 0 0:044 0:25 0:041 5 0 0 0:041 0:12 The elemental stiffness and mass matrices are assembled for cracked spring as given below. 2 3 16408:68 9082:8 0 0 6 9082:8 16021:44 6938:64 7 0 7 K ¼ 6 4 0 6938:64 18988:92 12050:28 5 0 12050:28 12050:28 2 0 3 0:265 0:046 0 0 6 0:046 0:242 0:038 0 7 7 M ¼ 6 4 0 0:038 0:23 0:041 5 0 0 0:041 0:12 Determination of Eigenvalues and Eigenvectors by MATLAB Programming The eigenvalues and eigenvectors are computed by MATLAB programming. Table 2 highlights the natural frequencies and corresponding mode shapes for healthy and cracked leaf spring. Determination of Dynamic Flexibility of Healthy and Cracked Spring Flexibility matrices of healthy and cracked springs are calculated after mass normalization. Table 2 Modal frequencies of healthy and cracked spring Mode shape Healthy spring (Hz) Cracked spring (Hz) First 61.95 Second 200 198.74 Third Fourth 346.70 502.72 333.07 467.1 123 58.01 3 0:021 0:026 0:026 0:026 6 0:026 0:043 0:048 0:048 7 4 7 ½Fhx ¼ 6 4 0:02 0:048 0:06 0:065 5  10 0:026 0:048 0:065 0:072 3 2 0:019 0:023 0:021 0:021 6 0:023 0:037 0:04 0:039 7 6 7  104 ½F dx ¼ 4 0:021 0:04 0:052 0:06 5 0:021 0:04 0:064 0:07 Dynamic Flexibility Change for Healthy and Cracked Springs The change in flexibility among both the spring is given below. D½F  ¼ 2 ½F hx ½F dx 0:019 0:023 0:021 6 0:023 0:037 0:04 ¼6 4 0:021 0:04 0:052 0:021 0:04 0:064 3 0:021 0:039 7 7104 0:06 5 0:07 The increase in flexibility in corresponding element indicates the presence and location of crack. Modal Curvature Method The location of crack is found on the modeled healthy and cracked cantilever beams after mass normalization. The mass normalized eigenvectors of healthy beam are as follows. 2 3 2 3 2 3 0:074 0:154 0:154 6 0:125 7 6 0:081 7 6 0:081 7 7 6 7 6 7 Uh1 ¼ 6 4 0:151 5 Uh2 ¼ 4 0:076 5 Uh2 ¼ 4 0:076 5 0:159 0:142 0:142 2 3 0:104 6 0:109 7 7 Uh3 ¼ 6 4 0:015 5 0:107 The mass normalized eigenvectors of cracked beam are as follows. 2 3 2 3 2 3 0:065 0:150 0:120 6 0:110 7 6 0:110 7 6 0:12 7 6 6 7 7 7 Ud1 ¼ 6 4 0:148 5 Ud2 ¼ 4 0:08 5 Ud3 ¼ 4 0:011 5 0:156 0:14 0:087 2 3 0:014 6 0:033 7 7 Ud4 ¼ 6 4 0:068 5 0:09 The modal curvature of healthy and cracked beam for all four nodal points can be calculated through central difference approximation. J Fail. Anal. and Preven. 2 3 2 3 1:042 15:76 Kh1 ¼ 4 01:73 5 Kh2 ¼ 4 05:83 5 01:25 2 3 26:32 3 22:01 5:486 Kh3 ¼ 4 21:32 5 Kh4 ¼ 4 12:36 5 01:94 19:58 2 3 2 3 2 3 1:38 13:19 25:0 Kd1 ¼ 4 0:486 5 Kd2 ¼ 4 10:42 5 Kd3 ¼ 4 24:24 5 2:08 3 09:03 0:76 2 4:23 Kd4 ¼ 4 10:28 5 17:99 The absolute modal curvature difference among the healthy and cracked beam approximates the location of crack for all four modes are given below 2 3 0:338 D½K 1 ¼ ½Kd1  Kh1  ¼ 4 1:244 5 0:83 2 3 2:57 D½K 2 ¼ ½Kd2  Kh2  ¼ 4 4:59 5 2:71 2 3 2:99 D½K 3 ¼ ½Kd3  Kh3  ¼ 4 02:92 5 02:7 2 3 1:256 D½K 4 ¼ ½Kd4  Kh4  ¼ 4 02:08 5 1:59 Fig. 5 Healthy leaf spring where D[K]1, D[K]2, D[K]3 and D[K]4 are the absolute modal curvature differences for the first four mode shapes. Results and Discussion The presence, location and severity of crack in composite leaf spring are evaluated as follows. Presence of Crack In order to confirm the presence of crack at a global level, the modal natural frequency is the suitable parameter. The comparisons of natural frequencies of cracked spring with the healthy for lower modes are made. It has been found that natural frequency is decreased for cracked spring. Figures 5 and 6 show frequency variation with mode number for healthy and cracked leaf springs obtained by analytical, ANSYS and experimental approaches, respectively. Figure 7 shows frequency change among both the spring by analytical and ANSYS results. Fig. 6 Cracked leaf spring Location of Crack Modal flexibility change and modal curvature difference parameters are used to locate the position of the intentionally created crack on the mono composite leaf spring using modal techniques. The modal flexibility change was found to be maximum between the portions of the beam 0.3–0.4 m as shown in Fig. 8. Similarly, the modal curvature difference was observed to be maximum between the portions of the beam 0.35–0.40 m as shown in Fig. 9. 123 J Fail. Anal. and Preven. Fig. 7 Change in frequency among both springs Fig. 9 Modal curvature method Fig. 8 Modal flexibility method Severity of Crack The severity of crack is quantified in terms of modal parameters such as modal flexibility and modal curvatures. The severity of crack for modal flexibility is observed to be 15.38% as shown in Fig. 10. Similarly, severity of crack for modal curvature found to be is 255.96 and 44.04% for 1st and 2nd modes, respectively, as shown in Fig. 11. Conclusion Failure analysis of composite mono leaf spring has been carried out by modal flexibility and curvature mode shape methods. The reduction in natural frequency showed the presence of crack in the spring at global level. The modal 123 Fig. 10 Modal flexibility method flexibility change was observed in the 3rd element at the crack location. Similarly, the modal curvature change is also observed in the same element at the crack region. The curvature mode shape method is more precise in locating the crack than the modal flexibility method. Thus, the crack location can be detected using two independent methods. The severity of crack was observed at 7th nodal point by both the methods. The maximum severity was found at 1st mode due to maximum amplitude of beam in the vicinity of crack region. The amplitude of mode shape goes on decreasing with the higher modes resulting in less severity at the crack region. J Fail. Anal. and Preven. Fig. 11 Modal curvature method References 1. M. He, Y. Tao Yang, Nondestructive identification of composite beams damage based on the curvature mode difference. J. Compos. Struct. (2017). https://doi.org/10.1016/j.comp.struct. 2017.05.040 2. X. Zhoua, D. Wangb, M. Duana, J. Gua, Y. Liuaa, Numerical study on mode curvature for damage detection of a drilling riser using transfer matrix technique. J. Appl. Ocean Res. 2017(63), 65–75 (2017) 3. Z.-B. Yang, M. Radzienski, P. Kudela, W. 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