Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 57, 1, pp. 49-58, Warsaw 2019 DOI: 10.15632/jtam-pl.57.1.49 INVESTIGATION OF FLEXIBILITY CONSTANTS FOR A MULTI-SPRING MODEL: A SOLUTION FOR BUCKLING OF CRACKED MICRO/NANOBEAMS Majid Akbarzadeh Khorshidi, Mahmoud Shariati Ferdowsi University of Mashhad, Department of Mechanical Engineering, Mashhad, Iran e-mail: mshariati44@um.ac.ir In this paper, a multi-springmodel is used for modelling of the crack in amicro/nanobeam under axial compressive load based on a modified couple stress theory. This model inc- ludes an equivalent rotational spring to transmit the bending moment and an equivalent longitudinal spring to transmit the axial force through the cracked section, which leads to promotion of the modelling of discontinuities due to the presence of the crack. Moreover, this study considers coupled effects between the bending moment and axial force on the discontinuities at the cracked section.Therefore, four flexibility constants appear in the con- tinuity conditions. In this paper, these four constants are obtained as a function of crack depth, separately. This modelling is employed to solve the buckling problem of cracked micro/nanobeams using a close-form method, Euler-Bernoulli theory and simply suppor- ted boundary conditions. Finally, the effects of flexibility constants, crack depth and crack location on the critical buckling load are studied. Keywords: flexibility constants, multi-spring model, MCST, buckling, crack 1. Introduction It is clear that presence of cracks or any other defects into any structure leads to a decrease of its capabilities. The issue of cracking in the structures is interested in both macro and small scale dimensions.Thus,presentation of anaccurate andappropriatemodel to capture crack conditions is very important. In many studies, cracks have been modeled by means of different types of springs. The type of the springmodel depends on problem type, such as the type of loading and geometry. In fact, kinds of displacements at the cracked section determinewhatmodelling should be selected. For example, a longitudinal spring model is used when the axial displacement is dominant (Hsu et al., 2011), a rotational springmodel is applicable for awide range of problems in which the angle changes between the crack surfaces are important (Akbarzadeh Khorshidi et al., 2017; Akbarzadeh Khorshidi and Shariati, 2017b; Hasheminejad et al., 2011; Ke et al., 2009; Loya etal, 2006; Shaat et al., 2016; Torabi and Nafar Dastegerdi, 2012; Wang andWang, 2013;Yang andCheng, 2008). Structures under torsion incorporate a torsional spring to describe discontinuity at the cracked section (Loya et al., 2014). Rice and Levy (1972) stated that the presence of a crack leads to a local reduction in bending and extensional stiffness along the crack line. Therefore, it is more accurate to use a model which considers these two local reductions. Akbarzadeh Khorshidi and Shariati (2017a) presented buckling analysis of cracked nanobeams based on a modified couple stress theory and using a two-spring model at the cracked section. The authors used thementionedmodel according to the historical background expressed byRice and Levy (1972) and the discontinuity relations presented by Loya et al. (2009). In majority of recent studies on static and dynamic behavior of micro/nanobeams in the presence of a micro or nano-scale crack, the flexibility constant which introduces the crack 50 M. AkbarzadehKhorshidi, M. Shariati severity is considered as a hypothetical input. However, there are studies which formulate the severity of the crack as a function of the crack depth, the material length scale parameter and other mechanical characteristics of the beam (Shaat et al., 2016; Akbarzadeh Khorshidi et al., 2017). Thesepapers use energy stored in the springand compare itwith the strain energy release rate at the crack surfaces. In the present study, the two-spring model is employed to describe discontinuities at the cracked section and, consequently, four flexibility constants appear, which gives the severity of the crack. Each flexibility constant is presented as a function of crack depth (as an unknown parameter) and other parameters (given values). Therefore, the continuity relations are formu- lated against the crack depth. Themacroscopic fracturemechanics is used formicro/nano-scale beams based on atomistic simulation models and continuum models (Joshi et al., 2010; Tsai et al., 2016; Hu et al., 2017). Then, amodified couple stress based solution is proposed for buckling analysis of the cracked beams. 2. Modelling Consider an Euler-Bernoulli beamwith lengthL, width b, thickness h and a crack with depth a is located at distanceLc from the left side of the beam (Fig. 1a). In the present modelling, the cracked beam is modelled as two separate segments connected by two massless elastic longitu- dinal and rotational springs (Fig. 1b). Therefore, the total strain energy of the cracked beam is equal to the strain energy of these two segments plus the strain energy stored in the springs. With this explanation, the released potential energy due to the presence of the crack is equal to the strain energy stored by the springs. The continuity conditions governed between the two beam segments are defined as follows (Akbarzadeh Khorshidi and Shariati, 2017a; Loya et al., 2009) w1 =w2 N1 =N2 M1 =M2 x=Lc ∆θ=KMMM+KMNN ∆u=KNNN+KNMM (2.1) where∆θ is the difference in the rotation angles between two crack surfaces (or the angle rotated by the rotational spring) and∆u is the longitudinal displacement occurredat the cracked section (or amount of longitudinal spring compression).N andM are, respectively, the axial force and the bendingmoment. Also,KMM,KMN,KNN andKNM are four coefficients to represent the coupled effects between the axial force and bendingmoment in discontinuity relations. Fig. 1. (a) A schematic of the cracked beam and (b) the springs model for a cracked section Therefore, the strain energy of springsUsprings is stated as Usprings = 1 2 M∆θ+ 1 2 N∆u= 1 2 M(KMMM+KMNN)+ 1 2 N(KNMM+KNNN) (2.2) Investigation of flexibility constants for a multi-spring model... 51 Based on generalized Irwin’s (Irwin, 1960) relation, the potential energy-release rate G is introduced as (Rice and Levy, 1972) G= (1−ν2)a E (πσ2bY 2 b +2 √ πσtσbYtYb+σ 2 tY 2 t ) (2.3) whereE is Young’s modulus, ν is Poisson’s ratio, σ and Y , respectively, reflect the stress and a dimensionless function of the crack depth to thickness ratio a= a/h. Indices t and b represent the status of the parameters in tension and bending, respectively. When a cracked beam is subjected to compression, it senses a local compliance at the crac- ked section, and the zones around the crack tend to open the crack lips. Based on the stress concentration at the crack tip, a uniform stress field distributes along the beam thickness (see AkbarzadehKhorshidi and Shariati, 2017b). Therefore, the crack lips suffer stretching and ben- ding (Fig. 2). Thebending stress and tensile stress (thickness average stress) defined inEq. (2.3) are shown as σb = Mh 2I = 6M bh2 σt = N A = N bh (2.4) where I in Eq. (2.4)1 represents the moment of inertia and, for a rectangular cross section, is equal to bh3/12. Also, A in Eq. (2.4)2 denotes the cross section area and, for the mentioned cross section, is equal to bh. Fig. 2. The stress field due to the applied load andmoment along the thickness The strain energy due to the presence of the crack is obtained as Uc = a ∫ 0 GdAc = b a ∫ 0 Gda (2.5) Substituting Eqs. (2.3) and (2.4) into Eq. (2.5), we have Uc = (1−ν2) Eb (36πM2 h2 a ∫ 0 aY 2b da+ 12 √ πMN h a ∫ 0 aYtYb da+N 2 a ∫ 0 aY 2t da ) (2.6) where a = a/h introduces the crack depth to thickness ratio. The dimensionless function Yt is defined as (Gross and Srawley, 1965) Yt =1.99−0.41a+18.70a2−38.48a3+53.85a4 (2.7) Also, the dimensionless function Yb is defined as (Ke et al., 2009) Yb =1.15−1.662a+21.667a2−192.451a3+909.375a4−2124.310a5 +2395.830a6−1031.750a7 (2.8) 52 M. AkbarzadehKhorshidi, M. Shariati We consider thatUspring represented in Eq. (2.2) is equal toUc obtained in Eq. (2.6), so, the flexibility constants KMM,KMN,KNN andKNM are separately achieved as follows KMM = 72π(1−ν2) Ebh2 a ∫ 0 aY 2b da KMN =KNM = 12 √ π(1−ν2) Ebh a ∫ 0 aYbYt da KNN = 2(1−ν2) Eb a ∫ 0 aY 2t da (2.9) Aswe know, the stress resultants introduced inEqs. (2.1) and (2.2) (the bendingmomentM and the axial forceN) are defined as N = ∫ A σxx dA M =M1+M2 = ∫ A zσxx A+ ∫ A mxy dA (2.10) whereM1 is the conventional bending moment and M2 is the couple moment that comes from themodified couple stress theory proposed byYang et al. (2002). The displacement field for the Euler-Bernoulli beam is u1 =u(x)−z dw dx u2 =0 u3 =w(x) (2.11) where u andw are the axial and lateral displacements of the midplane, respectively. Therefore, the nonzero strains and stresses are shown as εxx = du1 dx = du dx −z d2w dx2 σxx =Eεxx =E (du dx −z d2w dx2 ) (2.12) Also, the nonzero terms of the symmetric curvature tensor χ and the deviatoric part of the couple stress tensor m are defined as (Akbarzadeh Khorshidi and Shariati, 2017a; Yang et al., 2002) χxy =− 1 2 d2w dx2 mxy =−ℓ2µ d2w dx2 (2.13) These tensors consider the couple stress effects in themodified couple stress theory, and ℓ is a material length scale parameter to capture the size effect (Yang et al., 2002). µ=E/(2+2ν) is the shear modulus. Now, substituting Eqs. (2.12) into Eq. (2.10), we have N =EA du dx M =−(EI+ ℓ2GA) dw2 dx2 (2.14) where Deff =EI+ ℓ 2GA is the effective beam stiffness obtained based on the modified couple stress theory. According to Eq. (2.14), we can rewrite Eq. (2.1) as w1 =w2 du1 dx = du2 dx d2w1 dx2 = d2w2 dx2 x=Lc dw2 dx − dw1 dx =KMM d2w dx2 +KMN du dx u2−u1 =KNN du dx +KNM d2w dx2 (2.15) where KMM =DeffKMM KMN =EAKMN KNM =DeffKNM KNN =EAKNN Investigation of flexibility constants for a multi-spring model... 53 Thus, we have KMM =36π(1−ν)h [1+ν 6 + (ℓ h )2] a ∫ 0 aY 2b da KMN =12 √ π(1−ν2) a ∫ 0 aYbYt da KNM =6 √ π(1−ν)h2 [1+ν 6 + (ℓ h )2] a ∫ 0 aYbYt da KNN =2(1−ν2)h a ∫ 0 aY 2t da (2.16) Using Eqs. (2.7) and (2.8), and integrating from Eqs. (2.16), the flexibility constants are obtained as functions of the crack depth to thickness ratio, and they are represented as follows KMM =36π(1−ν)h [1+ν 6 + (ℓ h )2] a2(0.6612−1.2742a+13.1490a2−102.9316a3 +533.4547a4−2321.1924a5+11126.9823a6−50267.9855a7+175186.4492a8 −4399132.5842a9+772269.2856a10−927343.5821a11+723108.2196a12 −329586.3470a13+66531.7539a14) KMN =12 √ π(1−ν2)a2(1.1442−1.2596a+16.3259a2−93.4384a3+403.2692a4 −1303.1856a5+3902.0329a6−9790.7006a7+17593.8331a8−20534.4869a9 +14059.7654a10−4273.8259a11) KNM =6 √ π(1−ν)h2 [1+ν 6 + (ℓ h )2] a2(1.1442−1.2596a+16.3259a2 −93.4384a3+403.2692a4−1303.1856a5+3902.0329a6−9790.7006a7 +17593.8331a8−20534.4869a9+14059.7654a10−4273.8259a11) KNN =2(1−ν2)ha2(1.9800−0.5439a+18.6485a2−33.6968a3+99.2611a4 −211.9012a5+436.8375a6−460.4773a7+289.9822a8) (2.17) 3. Solutions According to theEuler-Bernoulli beam theory, the governing equations for buckling of a cracked micro/nanobeam are derived as (Akbarzadeh Khorshidi and Shariati, 2017b) (EI+ ℓ2GA) d4wi dx4i +P d2wi dx2i =0 { i=1 0¬x¬Lc i=2 Lc ¬x¬L d2ui dx2i =0 { i=1 0¬x¬Lc i=2 Lc ¬x¬L (3.1) Here the subscript i = 1,2 refers to the left and right segments of the cracked beam. The boundary conditions of a simply supported beam are expressed as u1(0)=w1(0)= 0 u2(L)=w2(L)= 0 d2w1 dx2 ∣ ∣ ∣ ∣ ∣ x=0 =0 d2w2 dx2 ∣ ∣ ∣ ∣ ∣ x=L =0 (3.2) 54 M. AkbarzadehKhorshidi, M. Shariati The general solution to Eqs. (3.1) can be obtained as wi(x)=Ai sin(αx)+Bicos(αx)+Cix+Di i=1,2 ui(x)=Fix+Hi i=1,2 (3.3) where α= √ P/Deff , Ai, Bi, Ci, Di, Fi and Hi are unknown constants to be determined from the boundary and continuity conditions. Applying continuity conditions (2.15) and boundary conditions (3.2) into Eqs. (3.3), the unknown constants are derived as A1 = ( 1− tan(αL) tan(αLc) ) A2 B1 =D1 =0 C1 =2α tan(αL) sin(αLc) L−Lc L A2 B2 =−tan(αL)A2 C2 =−2α tan(αL) sin(αLc) Lc L A2 D2 =2αLc tan(αL) sin(αLc) A2 F1 =F2 H1 =0 H2 =−LF2 F2 = A2α KMN ( KMMα[sin(αLc)− tan(αL)cos(αLc)]− tan(αL) sin(αLc) ) (3.4) also α= L+KNN KMNKNM KMMα ( sin2(αLc)− 12 tan(αL)sin(2αLc) ) − tan(αL) sin2(αLc)− 12 tan(αL)sin(2αLc) (3.5) The critical buckling load can be obtained by solving Eq. (3.5). For example, when we have an intact beam (a=0→KMM =KMN =KNN =KNM =0), according to Eq. (3.5) we have tan(αL)= 0 → α= nπ L n=1−→ Pcr =Deff (π L )2 (3.6) This is quite similar to the results obtained by Mohammad-Abadi and Daneshmehr (2014) for modified couple stress based intact microbeams. Using Eq. (3.5), the critical buckling load corresponding to each crack depth and crack location can bedetermined.Also, the presentmodel (four flexibility constants) can be compared with the common model (only one constant) by removing the other constants. Moreover, the coupled effects between the bendingmoment and axial force can be evaluated by neglecting the crossover flexibility constants (KMN andKNM). Thus, the following equation canbeusedwhenonlyoneflexibility constantKMM is employed KMMLα ( sin2(αLc)− 1 2 tan(αL)sin(2αLc) ) − tan(αL) =0 (3.7) Also, the following equation can be used when the crossover flexibility constants are removed (L+KNN)KMMα ( sin2(αLc)− 1 2 tan(αL)sin(2αLc) ) − tan(αL)= 0 (3.8) 4. Results To illustrate the flexibility constants effects on the buckling behavior of cracked mi- cro/nanobeams, some numerical examples of the obtained solution are presented in this Sec- tion. Also, the effects of the crack depth and crack location on the critical buckling load are investigated. First, the obtained results are validated with (Ke et al., 2009; Wang and Quek, Investigation of flexibility constants for a multi-spring model... 55 2005) for macro-scale cracked beams (ℓ = 0). This comparison can be observed in Table 1, so that P =Pcr/Pcr0 is the nondimensional critical buckling load (where Pcr0 denotes the critical buckling load of an intact beam). In (Ke et al., 2009;Wang andQuek, 2005), only one flexibility constant KMM (one equivalent rotational spring model) was employed, so, the present results have two separate columns for the one-spring model where we have only KMM and the two- spring model where all flexibility constants appear. In Table 1, each crack depth corresponds to the crack severity, for example, a/h = 0.1 corresponds to KMM = 0.01 (this parameter is introduced with symbolΘ in (Ke et al., 2009)). Table 1. Nondimensional critical buckling load P of a cracked beam (Lc = 0.5L, ν = 0.33, E =70GPa andL=10h) a/h Present Ke et al. Wang &Quek Two springs One spring (2009) (2005) 0.1000 0.9802 0.9801 0.9809 0.9830 0.1425 0.9614 0.9611 0.9622 0.9630 0.1757 0.9432 0.9426 0.9442 0.9450 0.2038 0.9257 0.9245 0.9266 0.9250 0.2280 0.9092 0.9071 0.9096 0.9070 Now, Table 2 and Figs. 3-6 present the critical buckling load for cracked micro/nanobeams based on themodified couple stress theory and the two-springmodel. All results are obtained as a parametric studywhere ν =0.33,L/h=10 and ℓ/h=0.5. The present study is applicable for bothmicro and nano-scale problems (this issue is dependent on the scale of thematerial length scale parameter). Table 2 presents nondimensional critical buckling loads for different crack depths. In this table, three types of nondimensional critical buckling loads are shown, where each load denotes a special case of the flexibility field. P1 is the nondimensional critical buckling load for the one-spring model where we have only KMM (conventional model), P2 is for the two-spring model, but the crossover flexibility constants are vanished (the coupled effects between the axial force and bending moment are neglected) and P3 is for the two-spring model where all four flexibility constants are considered. The results of Table 2 indicate that there are some differences betweenP3 andP1, and this discrepancy increaseswhen the crack depth is increased. Also, comparison between P2 and P1 reveals that the use of two springs without consideration of the crossover constants has no considerable impact on the obtained results. Figure 3 approves Table 2, graphically. It is found that the two-spring model presents a greater buckling capacity of cracked beams than the conventional model. Therefore, it is found that the local flexibility at the cracked section (crack severity) caused by a particular crack depth is different for the one-springmodel P1 and the two-spring model P3. Table 2.Nondimensional critical buckling load P of cracked micro/nanobeams (Lc =0.5L) a/h P1 P2 P3 0 1 1 1 0.1 0.9584 0.9584 0.9586 0.2 0.8545 0.8545 0.8567 0.3 0.7130 0.7130 0.7232 0.4 0.5489 0.5489 0.5772 0.5 0.3901 0.3901 0.4504 56 M. AkbarzadehKhorshidi, M. Shariati Fig. 3. Comparison of the two-springmodel and the conventional model in terms of crack depth The effect of crack location is shown inFig. 4 for different crack depths. This figure indicates that the crack has the greatest sensitivity when it is located inmiddle of the beam (Lc =0.5L). When the crack approaches the two ends, its effect is continuously decreased.This fact is directly related to deformation of various points of the beam and, finally, the opening of the crack tip. Fig. 4. The effect of crack location on the nondimensional critical buckling load with different crack depths Investigation of flexibility constants for a multi-spring model... 57 Also, variations of the nondimensional critical buckling load versus crack depth are demon- strated in Fig. 5 in different crack locations. It is observed that not only the increasing of the crack depth leads to a decrease in the buckling resistance of the beam, but alsomakes the effect of crack location more considerable. Fig. 5. The effect of crack depth on the nondimensional critical buckling load with different crack locations 5. Conclusion The flexibility constants of the cracked section are investigated using a multi-spring model (ro- tational and longitudinal spring) to describe local flexibilities and discontinuities at the cracked section ofmicro/nanobeams.Thismodel not onlypromotes thediscontinuities but also considers the coupled effects between the bending moment and axial force on the discontinuities due to the presence of the crack. Then, the buckling problem is solved for cracked micro/nanobeams and the influence of crack depth and crack location is studied. Also, different configurations of the flexibility constants are compared together. The results show that the flexibility constant related with the bendingmoment (KMM) has the greatest impact on the local flexibility due to the crack (crack severity). But, this crack severity changes by addingmore flexibility constants. It is found that the coupled effects between the bendingmoment and axial force (crossover con- stants) are considerable, and the making use of the multi-spring model without consideration of the crossover constants will not be useful. Therefore, the use of four constants (multi-spring model) instead of only one (conventionalmodel) estimates the buckling capacity better, and this difference increases with an increase in the crack depth. References 1. AkbarzadehKhorshidiM., ShaatM.,AbdelkefiA., ShariatiM., 2017,Nonlocalmodeling and buckling features of cracked nanobeams with von Karman nonlinearity, Applied Physics A: Material Science and Processing, 123, 62 58 M. AkbarzadehKhorshidi, M. Shariati 2. Akbarzadeh Khorshidi M., Shariati M., 2017a, A multi-spring model for buckling analysis of cracked Timoshenko nanobeams based onmodified couple stress theory, Journal of Theoretical and Applied Mechanics, 55, 4, 1127-1139 3. 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