Effects of contact between rough surfaces on the dynamic responses of bolted composite joints: Multiscale modeling and numerical simulation

Composite Structures 211 (2019) 13–23

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Effects of contact between rough surfaces on the dynamic responses of bolted composite joints: Multiscale modeling and numerical simulation

T

Zhen Zhanga,b, Yi Xiaoa, , Yuanhong Xiea, Zhongqing Sub ⁎

a b

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, PR China Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong Special Administrative Region

ARTICLE INFO

ABSTRACT

Keywords: Bolted joint Rough interface Damping ratio Fractal analysis Composite structures

A multiscale numerical model, considering both microscopic interfacial properties and macroscopic composite properties, has been developed to model the dynamic responses of bolted composite structures using the elasticviscoelastic correspondence principle. The complex contact moduli of the uneven interfaces of the joints, featuring multi-asperity contact at the micro perspective, were derived using the fractal contact theory. The frequency-dependent complex moduli of composite materials were ascertained through modal impact tests on unidirectional composites. The damping and stiffness of the composite specimens were solved within ABAQUS software using the complex eigenvalue method. The damping ratios of the joints decreased by over 8.5% while the resonant frequencies increased by more than 0.59% when the bolts of the joints were adjusted from fully loose to fully tightened. Results from the numerical prediction and the experiment were found to be in good agreement, whereby the influence of the interfacial contact conditions on the dynamic responses of the bolted composite structures was revealed through the multiscale analysis.

1. Introduction

dynamic characteristics of composite structures have been successfully predicted. However, four main challenges still exist that must be solved prior to achieving an accurate prediction of the modal parameters of bolted composite structures during vibration fatigue: (1) theoretical derivation of the stiffness and damping properties for rough contact surfaces; (2) implementation of numerical modeling for the contact properties of rough interfaces; (3) implementation of numerical modelling of the frequency-dependent damping properties for the composite materials; (4) prediction of the evolution of stiffness and damping properties of rough contact surfaces produced by vibration fatigue loadings. Rough surfaces that are a result of manufacturing tolerances usually lead to complex energy dissipation behaviors of a bolted joint [9–11]. As a result, understanding the contact mechanisms at the rough surfaces is still a challenging problem, and inaccurate modeling of this may cause unacceptable errors during the dynamic analysis of bolted composite structures. Without loss of generality, the joints represent discontinuities in a bolted structure and may lead to high-stress concentrations and further structural failure. It has been reported that for a bolted metallic structure, up to 90% of the structural damping is generated by the joint interfaces which is usually designed to limit vibration amplitude [12], and around 50% of the system stiffness is contributed by the joint interfaces [13,14]. Therefore, the basic challenge

Advanced composite materials have been widely adopted in various industrial fields for weight critical applications such as aerospace, automobile, and marine engineering, due to their advantageous properties typified by high strength-weight ratio and high stiffness-to-weight ratio [1]. The preferred method of assembling composite structures is to use bolted joints, in order to streamline manufacturing and facilitate system maintenance [2–5]. However, lightweight structures, including bolted composite joints, are prone to vibration, which may result in unexpected instability (i.e., intensive vibration arising from resonance [3]) and failure modes (e.g., interfacial wearing, fiber and resin breaking and delamination [5]) and can reduce structural integrity [4,6]. Therefore damping capability plays a critical role in determining the long-term performance of a bolted composite structure subject to vibration [3,7]. The authors’ previous work revealed that the damping ratios of bolted composite joints increased and resonance frequency decreased due to interfacial wear and material internal damage when the joints were subject to vibration fatigue [3]. Therefore, it is of vital importance to develop an accurate model to predict the dynamic properties of bolted composite structures, including the damping ratio and the resonant frequency [8]. With the development of numerical techniques, the static and ⁎

Corresponding author. E-mail address: [email protected] (Y. Xiao).

https://doi.org/10.1016/j.compstruct.2018.12.019 Received 25 December 2017; Received in revised form 22 October 2018; Accepted 11 December 2018 Available online 12 December 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved.

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Nomenclature

Syz ,Szx a al as p pp

tangential compliance of the rough interface truncated asperity contact area largest truncated area smallest truncated area normal load carried by a single-asperity contact plastic asperity contact load equivalent plastic normal modulus of single asperity gp equivalent plastic shear modulus of single asperity asperity interference w input energy induced by the asperity tangential force R asperity radius r radius of truncated area E equivalent Young’s modulus of two contact surfaces G equivalent shear modulus of the two surfaces Gt shear modulus of the whole interface Gt complex shear modulus of the rough interface μ friction coefficient Poisson’s ratio of two contact surfaces 1, 2 G fractal roughness equivalent loss factor loss factor of the joint interface c T applied torque P normal contact load of the whole interface W input energy induced by the interfacial tangential force [K Interface] interfacial stiffness matrix [SInterface] interfacial compliance matrix Szz normal compliance of the rough interface

asperity contact area largest asperity contact area smallest asperity contact area critical asperity contact area elastic asperity contact load equivalent elastic normal modulus of single asperity equivalent elastic shear modulus of single asperity tangential load carried by a single-asperity contact dissipated energy arising from the asperity microslip size distribution of asperity truncated area radius of contact area Young’s modulus of two contact surfaces shear modulus of two surfaces Young’s modulus of the whole interface complex Young’s modulus of the rough interface hardness of the softer material of the two in contact equivalent Poisson’s ratio fractal dimension fractal domain extension factor loss factor of fasteners f damping ratios Ft tangential load carried by the whole interface Wd dissipated energy arising from the interfacial microslip A0 nominal contact area of the rough surface [K Composite] complex composite stiffness matrix

a al as ac pe ee ge t wd n (a ) r E1, E2 G1 G2 En En H ve D

for accurately predicting the interfacial energy dissipation and the structural stiffness of a bolted joint is to develop an efficient model that is capable of describing rough surface interactions within the bolted structure [8]. Fractal theory can characterize rough surfaces without dependence on the resolution and sampling length of the measuring instruments. Majumdar and Bhushan’s (MB) fractal theory [15] has been widely adopted to derive the contact stiffness and damping of the rough interfaces of joints. For instance, Wang et al. [16] characterized contact surfaces as fractals and used fractal geometry to analyze temperature rises at the microcontacts in a slow sliding region. Zhang et al. [17] proposed a modified MB fractal model to investigate the mechanisms of energy dissipation at joint interfaces and further derived the tangential stiffness and damping loss factor of the joint interfaces. Zhao [14] established a modified three-dimensional fractal model to analyze the stiffness and damping of rough contact surfaces for a bolted metallic joint. Three predominant methods are currently used to model the contact properties of rough interfaces in finite element analysis [18], namely node-to-node contact, thin-layer elements, and zero-thickness elements. Of these methods, the virtual material model, using thin-layer elements, can be easily integrated with commercial finite element software including ABAQUS [13,19–22]. For instance, Tian et al. [13] conducted virtual material-based numerical modeling of fixed joint interfaces and successfully improved the modeling accuracy of a machine tool that incorporated joints. Two prevailing approaches have been used to model damping properties of composite materials, namely the strain energy approach and the elastic-viscoelastic correspondence principle [23]. However, the frequency dependence or the viscoelastic properties of the composite materials are usually ignored in the former method. The authors have previously adopted the latter method in the research [24] to provide an efficient prediction of the modal damping properties of anisotropic materials using the ABAQUS code. In this study, a multi-scale numerical model developed in the ABAQUS software, consisting of composite materials, thin-layer

interfaces (virtual materials), and fasteners, has been proposed in order to quantitatively investigate the influence of microscale surface contact on the dynamic structural responses of a bolted composite joint using the elastic-viscoelastic correspondence principle. The research flowchart is shown in Fig. 1. The complex moduli of the rough contact surfaces of the joint have been quantitatively linked to the contact pressure, derived by using the fractal contact theory in conjunction with microscopic observation. The frequency-dependent complex moduli of the composite materials were ascertained from modal tests on unidirectional composites. Subsequently, the damping and stiffness of the bolted composite structures were modeled by using the complex moduli of both the composite materials and the rough interfaces. The dynamic responses of the bolted composite joints with different leftover torque values were solved using the complex eigenvalue method in ABAQUS, whereby a quantitative relationship between the modal parameters of the joints and the applied torque was built. Modal impact tests were conducted on both cross-ply composite beams and bolted composite joints subject to different torques, to verify the efficiency of the proposed numerical analysis. The results calculated from numerical models with rough interfaces and perfect interfaces respectively were further compared in order to reveal the influence of rough interfacial contact on the dynamic responses of bolted composite structures. 2. Theoretical analysis of the contact stiffness and damping of rough surfaces In this study, virtual material model using thin-layer elements was used to simulate the contact properties of rough interfaces to obtain the dynamic responses for bolted composite joints under different torques. The surfaces of composites are usually uneven at the micro perspective and have randomly distributed resin asperities, and the rough surface profile obtained from a scanning electron microscope (SEM) is shown in Fig. 2. The results obtained from the metallographic tests (MGT) showed that a thin resin layer (approximately 5 µm ), with randomly distributed asperities, covered the surface of the composites. Therefore, in the numerical model, 5 µm thin isotropic layers were adopted to 14

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Fig. 1. Research flowchart.

model rough resin layers on the composite surface. The interfacial material properties (i.e., complex moduli) of the rough surfaces under different pressure induced by the applied torques will be derived using fractal contact theory. Following assumptions of the resin rough surface were adopted to simplify the analysis [13,17]: (1) the rough surface is isotropic; (2)

asperities on the rough surface are spherical near their summits with different radii; (3) asperities are far apart and no interaction in between; (4) asperities undergo elastic or fully plastic deformation and no hardening and bulk deformation; (5) partial sliding contact occurs between the interfaces and tangential energy dissipation dominates the interfacial damping; (6) tangential energy dissipation of sliding

Fig. 2. Schematic illustration of the setup for modal analysis of a bolted composite joint. 15

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asperities in a cycle is independent of vibration frequency. Note that the isotropic material assumption only holds true for the thin resin layer on the composite surface. According to the above assumption, two contacting rough surfaces can be simplified as an equivalent rough surface containing spherical asperities with different radii in contact with a smooth rigid plane. The interaction between a typical asperity under a normal concentrated force and a rigid plane is displayed in Fig. 3(b). For the spherical asperity tip, the resulting microcontact region after deformation is circular with a contact area of a (radius r) and a truncated area (intersection of the ‘undeformed’ sphere and the rigid plane) of a (radius r ) without deformation. In the fractal contact model, asperity contacts with different contact areas are hypothesized to undergo different degrees of deformation [15]. Large contacting asperities form contact areas (a ) that exceed the critical area (ac ) and therefore undergo elastic deformation. While asperities that have contact areas that are less than the critical area (ac ) are in plastic deformation. For asperities that undergo elastic deformation, the reliance of the contact load ( pe ) on the truncated contact area (a ) can be expressed as [17]:

pe = 4/[3(2 )0.5] E G D 1a

(3 D)/2

contacts takes the following form [16]

n (a ) = 0.5D

v12)/ E1 + (1

(D + 2)/2

(0 < a < a l )

(9)

Here a l represents the maximum truncated elastic area. denotes the domain extension factor and can be expressed with fractal dimension D as: (2 D)/2

(1 +

D /2)(D 2)/ D

= (2

D )/D

(10)

In the above analysis, the contact properties of the asperities were expressed with the truncated contact area, and the area distribution of a contact interface can be characterized with fractal parameters (i.e., D and G). Then based on statistical analysis, the normal contact load P of the whole interface, with multi-asperity contacts, can be obtained by using the truncated area a as an integral parameter to calculate the contact loads carried by single-asperity contacts in both elastic and plastic deformation. In other words, the total interfacial contact load P can be determined by integrating the contribution of the elastic (with truncated area a c < a < a l ) and plastic asperities (with truncated area a s < a < a c , a s represents the minimum truncated plastic area) as:

(1)

where D and G denote the fractal dimension and roughness, respectively. E is the equivalent Young’s modulus of two contact surfaces possessing Young’s modulus (E1 and E2 ) and Poisson’s ratio ( 1 and 2 ):

1/ E = (1

(2 D)/2a D/2 a l

(2)

v22 )/E2

Note that Eqs. (1) and (2) assume that contact materials are isotropic, which is only applicable for the thin resin layer on the composite surface. According to Hertzian contact theory, a sphere (radius R) in contact with a rigid plane forms a contact area a = R ( denotes the asperity interference) [25]. Therefore, the relation between the truncated contact area (a ) and the contact area (a ) in the elastic deformation range as shown in Fig. 3(b) can be expressed as following [13,26]:

a = r

2

(R2

=

(R

) 2) = 2 R = 2a

(3)

The critical truncated area (a c ) which distinguishes between elastic and plastic deformation of the asperities can be written as [17]:

a c = 2G 2/[(0.5HE ) 2/(D

(4)

1)]

where H is the hardness of the softer material of the two in contact. According to Hertzian contact theory and micro slip theory, a tangential force less than the limiting friction force causes a small relative motion between two contact asperities, referred to a micro-slip region over part of the interface. For the partial slipping contact, the equivalent normal (ee ) and shear modulus (ge ) of single-asperity contact in the elastic deformation can be expressed as [13]:

ee = 4/(3 ) E

R/ ; ge = 16/ [1

t /(µp)]1/3 G

(a)

(5)

Here p and t are the normal and tangential loads carried by a singleasperity contact. μ refers to the friction coefficient.

R = a D/2/(2 G D 1);

= G D 1a

(D 1)/2

(6)

G is the equivalent shear modulus of the two rough surfaces with shear moduli G1 and G2 , respectively, as:

1/ G = (2

v1 )/G1 + (2

v2 )/G2

(7)

While for asperities in the plastic deformation (to simplify the analysis, perfect plasticity was assumed for the asperities on the contact materials and hardening was neglected for the micron-sized asperities on the thin resin layer), the contact load pp and the contact moduli (plastic Young’s modulus , and shear modulus gp ) can be expressed with the truncated area as [13]:

pp = 0.5Ha ; ep = gp = 0

(b)

(8)

Fig. 3. (a) Elastic and plastic contact of asperities at a rough interface and (b) a spherical asperity in contact with a rigid flat surface.

The size distribution of the truncated area n (a ) of the asperity 16

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Z. Zhang et al. al ac

P=

pe n (a ) da c

a c as

+

pp n (a ) da

20.25/(

) EG 0.5

1 0.5D

=

+

0.25a 0.75 ln(a / a ) l c l

+ 6H

0.25a 0.75 a 0.25 l c

23 0.5DEDG (D 1) 0.5D 1.5 D al (al 3(3 2D)

ac1.5 D )

al

(D

/(2A0

1.5)

a l

Gt =

1 0.5D

[0.5ge n (a ) a ]da /A 0 = 16al0.5D (ac1

ac

/[A0 (D

0.5D

al1

0.5D

(13)

Ft /(µP )]1/3 G D

2)][1

yy zz xy

When a single-asperity contact is subject to both normal and tangential loads, both the adherence region and the microslip region exist at the whole contact surface in the case that the tangential load t is less than the limiting friction. It has been reported that microslip in the tangential direction caused by tangential force is the main mechanism of energy dissipation for rough contact interfaces [13,17,19] and energy dissipation from asperity plastic deformation and microslip caused by normal load was neglected in this study. The dissipated energy wd arising from the microslip, and the input energyw induced by the tangential force can be expressed as [17]:

[1

t /(µp)]5/3

w=3

r ){3/5 + 2/5[1

t /(µp)]5/3

[1

t /(µp)]2/3 } (17)

W=

a l a c

al ac

1

wd n (a ) da =

wn (a ) da =

9(2 ) 2 µ2 P 2 (2 10G

(2

D) 2al

(D 4)/2

D )2al

(D 4)/2

(al

(3 D)/2

(2 D)/2D (3

(ac (3

(2 D)/2D (3

D)/2

D)

al

(3 D)/2

ac (D

4)/2

)

D)

)

2

=

yy

Sxy

xy yz

Syz Szx

zx

(22)

(23)

A multi-scale numerical model, as shown in Fig. 4, was developed in the commercial numerical software ABAQUS to quantitatively investigate the effect of rough surface contact on the dynamic responses of a bolted composite joint, based on the previous fractal analysis of interfacial contact stiffness of rough surfaces and modal analysis of composite materials. 3.1. Microscopic analysis of the complex modulus of rough surfaces

Using fractal theory, the dissipated energy Wd and the input energy W of the whole interface can be integrated as: Wd =

(21)

3. Numerical analysis of the dynamic responses of bolted composite joints

(16)

µ2 a2p2 /(8Ar2 G

(20)

Szz = 1/ En ; Syz = Szx = 1/ Gt

t /(µp))2/3]}

5t /(6µp)[1 + (1

2 3

Here, Szz represents the normal compliance, while Syz and Szx define the tangential compliance of the rough interface. The Sxx , Syy and Sxy are zero because no stiffness in the directions parallel to the joint interface (Sxx and Syy ) or in-plane shearing (Sxy ). The off-diagonal terms (Sxy , Sxz and Syz ) in the matrix are also zero because there is no transversal contraction [18].

(15)

wd = 9µ2 p2 /(10rG ){1

2 3

zz

zx

The equivalent Poisson’s ratio of the rough interface can be calculated from [13]

1

Ft µP

xx

yz

(14)

ve = En/(2Gt )

Ft µP

1

Sxx Sxy Sxz Syy Syz Szz

xx

Ft is the tangential load carried by the whole interface and A0 is the nominal interfacial area. The relation between the loads (i.e., Ft and P ) carried by the whole interface with multi-asperity contacts, and the loads (i.e., tangential load t and normal load p ) carried by single-asperity contacts can be written as [17]: Ft / P = t / p

5 3

5Ft 1+ 1 6µP

Then the orthotropic constitutive matrix ([SInterface], refer to interfacial compliance matrix) of the virtual material representing the rough interface can be written in the following form as:

1 0.5D

)

Ft µP

5 3

En = En; Gt = Gt + i c Gt

(12)

ac0.5]

Ft µP

1

The elastic-viscoelastic correspondence principle was subsequently used to describe the stiffness (real part of the complex modulus) and the damping properties (imaginary part of the complex modulus) of the rough interface. The complex Young’s modulus En and the shear modulus Gt of the rough interface are written as:

(11)

2 ) EG1 Da D /2 [a 0.5 l l

3 2 1 + 5 5

/

2DH 0.5D 1 0.5D a ac 2 D l

[0.5ee n (a ) a ] da /A0 = 3D

ac

1 Wd = 1 2 W

(D = 1.5)

Young’s modulus En and the shear modulus Gt of the whole rough interface can be obtained as

En =

=

The surface roughness profile of the composite material was mea1

1

Ft µP

3(2 )1/2µ2 P 2 3 2 + 1 16G 5 5

5 3

5Ft 1+ 1 6µP

Ft µP

5 3

1

Ft µP

Ft µP

2 3

(18)

2 3

(19)

sured in order to obtain the interfacial properties of rough contact surfaces in bolted composite joints, as depicted in Fig. 2. The fractal dimension (D) and the fractal roughness parameter (G) were

The damping loss factor c of the interface can be expressed with the dissipated energy Wd and the input energy W during the microslip as

17

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Z. Zhang et al.

Fig. 4. Multiscale numerical model of a bolted composite joint with thin-layer interfaces.

subsequently ascertained as D = 1.22, G = 9.9852 × 10 9 through the power spectrum analysis [14] of the resin surfaces. Two types of contact interfaces, as shown in Fig. 4, were considered in the numerical model, namely the interface between the composite material and the metallic fastener (i.e., the M-C interface), along with the interface between the composites (i.e., the C-C interface). The interfacial properties (the normal storage modulus En , the tangential storage modulus Gt , and tangential loss factor c ) of the two interfaces were calculated using fractal contact analysis under different normal pressures P, induced by the applied torque T (subject to P = T /( d ) [27], where d and are the bolt diameter and the friction coefficient between the nut and bolt thread, respectively), as displayed in Fig. 5. The material properties of the T700/7901 composites and the steel fastener that were utilized for the simulation are shown in Table 1. There are four independent elastic constants: Young’s moduli in the longitudinal (E11) and transverse (E22 ) directions, the shear modulus (G12 ), and the major Poisson’s ratio ( 12 ), which were measured through tensile tests on 0°, 90° and 45° unidirectional composites. v23 and G23 were derived assuming transverse isotropy for tested composites using v23 = v12 (1 v12) and G23 = 0.5E22/(1 + 23) [28]. 12 E22 /E11)(1 Multiple energy dissipation mechanisms exist in the bolted composite joints including tangential microslip in the C-C and M-C interfaces, thread friction between bolt and nut, plastic deformation and material internal cracks, which introduce uncertainty in damping analysis of bolted composite joints. It is challenging to directly measure the energy dissipation caused by thread friction. Previously studies reported that around 50%–67% energy dissipation arises from the fasteners [3,29]. In order to consider the thread friction between the bolt and nut and simplify the analysis, the damping loss factor f , describing the energy dissipation between the fasteners, was assumed to be proportional to the applied torque. f was assumed being equal to c for T = 1 N∙m and T = 13 N∙m, which lead to an assumption around 50%–75% damping from fasteners in torque range from 1 to 13 N∙m. The equivalent loss factor , a sum of c and f ( = c + f ), was used to describe the interfacial energy dissipation of the joint. In other words, the thread friction damping was introduced in the virtual

(a)

(b) Fig. 5. Interfacial properties when the bolt is tightened with varied torques: (a) C-C interface and (b) M-C interface. 18

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Table 1 Material properties of the composites and the metallic fastener used in the numerical simulations. Material T700/7901 Steel

kg/m3 1700 7850

E11 GPa

E22 (E33 ) GPa

G12 (G13 ) GPa

G23 GPa

130 210

7 210

3.7 80.8

2.4 80.8

12 ( 13 )

23

0.32 0.30

0.45 0.30

Table 2 Experimental and numerical modal parameters of cross-ply composites. Specimen length

100 mm 130 mm

Resonant frequency [Hz]

Damping ratio [10−3]

Prediction

Test

Error

Prediction

Test

Error

221.64 131.28

218.58 127.27

1.40% 3.15%

0.923 0.775

0.902 0.800

2.34% −3.08%

(a)

(a)

(b) Fig. 6. (a) Frequency dependence of the damping ratios of unidirectional composites and (b) Damping ratio vs. maximum displacement of composites.

material interface rather than in the actual location (thread between the bolt and nut) to simplify the analysis. 3.2. Macroscopic analysis of the frequency-dependent complex modulus of composites

(b)

In this section, the complex moduli of composite material are ascertained using the same procedure in our previous work [24]. The storage moduli of the composites were measured in static tensile tests on the 0°, 45° and 90° unidirectional T700/7901 composites. The reliance of the composite loss moduli on the frequency was determined with the damping ratios ( ) of the above unidirectional composites of different lengths obtained from modal impact tests (to be detailed in Section 5), as displayed in Fig. 6. It was observed that the damping

Fig. 7. Damping ratio vs. maximum displacement of joints under different torques (a) J100 joint and (b) J130 joint.

ratios of the 45° ( 45° ) and 90° ( 90° ) unidirectional composites, which were dominated by the resin, were much higher than that of the 0° ( 0° ) unidirectional composite, which was dominated by the fiber. The frequency-dependent complex moduli of the composites consisting of 19

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J10 0

J1 00 Fitting curve

(a)

(b)

J 100

J10 0

(c)

(d)

Fig. 8. Experimental (a) displacement decay curves and (b) frequency spectra; numerical (c) displacement decay curves and (d) frequency spectra of the J100 joint under different torques.

storage and loss moduli, given as

= 2 [30], can be written as:

E11 = E11 + E

11

= E11 (1 +

11 i ) = E11 (1 +

E22 = E22 + E

22

= E22 (1 +

22 i )

= E22 (1 +

G12 = G12 + G

12

= G12 (1 +

12 i )

= G12 (1 +

° i)

(24)

90° i )

(25)

0

45° i )

element meshes and boundary conditions is shown in Fig. 4. The specimen analyzed included two laminated beams (joint materials), two types of thin-layer virtual interfaces and the metallic fasteners. The left end of the joint was fixed in place to form a cantilever beam. A tie was used in the ABAQUS model for the contact interfaces. An aspect ratio less than 100 was selected for the mesh of the virtual material (maximum ratio 99.97 and average ratio 92.7). The thin-layer virtual material, which possessed both a normal and tangential complex modulus and was defined by a UMAT subroutine for ABAQUS Standard, was used to simulate contact conditions of the M-C and C-C interfaces. When modeling the joint with different residual torques remained on the bolt, the complex normal and tangential moduli of the interfaces were adjusted according to the results shown in Fig. 5. The complex stiffness matrix of the composite materials was defined using the frequencydependent complex moduli measured from the experimental data as shown in Fig. 6. When the complex stiffness matrix of the virtual material interfaces and composite materials had been defined, the complex frequency analysis implemented in ABAQUS, was adopted to calculate the resonant frequency and damping ratio of the bolted composite joint subject to different applied torques. The displacement decay curves of the joint, when subjected to an impact force, were calculated using modal dynamic analysis.

(26)

It should be noted that 23 = 13 = 45° [31] was adopted in this study, therefore, G23 = G23 (1 + 45° i ) . Using the classical lamination theory, in conjunction with the elastic-viscoelastic correspondence principle, the complex stiffness matrix ([K Composite]) of laminated composites of different geometric sizes can be obtained. 3.3. Multiscale modeling of joint specimens The equation motion of a bolted composite joint with the inertia matrix ([MInterface]and[MComposite]) and complex stiffness matrix ([K Interface]and [K Composite]) of the rough interfaces and the composite materials can be written as

([MComposite] + [MInterface]){x¨} + ([K Composite] + [K Interface]){x } = {F }

(27)

Using the complex eigenvalue method, the resonant frequency ( ) and the loss factor ( s ) of the composite joint can be determined as:

([MComposite] + [MInterface]) = Re(

) ;

s

2

+ ([K Composite] + [KInterface]) = 0

= Im(

2

)/Re(

2

)

(28)

4. Experimental validation

(29)

In order to validate the numerical analysis with regards to the dynamic responses of the bolted composite joints with distinct leftover torque values, the vibration modal parameters including both the resonant frequency and the damping ratio were comparatively measured

For the present study, the bolted composite joint (see Fig. 2) was simulated as a 1/2 symmetry model using eight-node solid elements (C3D8) to reduce model size. A schematic representation of the finite20

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Z. Zhang et al.

J100

J100

(a)

(b) J130

J130

(c)

(d)

Fig. 9. (a) Damping ratio vs. applied torque and (b) resonant frequency vs. applied torque for the J100 joint; (c) damping ratio vs. applied torque and (d) resonant frequency vs. applied torque for the J130 joint.

from modal impact tests carried out on the joints. The beam specimens were cut from sheets of laminated T700/7901 carbon-fiber-reinforced epoxy, which were obtained using hot pressing with a stacking sequence [90, 0]4s. The specimens were 2 mm thick and 30 mm wide. Two composite beams with respective lengths of 100 mm and 130 mm, called C100 and C130 beams, were prepared for modal tests on the composite materials. Composite beams with a length of 50 mm were separately assembled with two beams with respective lengths of 70 and 100 mm, using an M6 bolt with a lap length of 20 mm, to form two bolted composite joints, termed as J100 and J130 joints. It should be noted that the total lengths of the C100 beam and J100 joint were the same and this was also the same for the C130 beam and J130 joint. The setup for the modal impact test is shown in Fig. 2, which consisted of test specimens, fixtures, an impact hammer, an eddy current displacement meter (Donghua 5E106), and a dynamic strain indicator (Donghua 5923N). The joint was clamped vertically and one end was secured (i.e., the beam with a length of 50 mm) by jaws of the fixtures. Impact forces in the thickness direction with different magnitudes were manually applied at different points near the clamped end on the specimens during the test to avoid node points for high-order modes (note: only 1st- order bending mode was investigated in the study). The displacement meter was used to measure the displacement responses of the specimens in the thickness direction with a sampling frequency of 10 kHz. The damping ratios were measured by fitting the displacement decay curves with the logarithmic decay method and the 1st-order resonant frequencies were determined in the frequency spectra. To minimize the influence of higher-order vibration modes which quickly decay in the free decay signal, the fitting interval was set when the response displacement was between 1/3 of the maximum signal

displacement and 0.05 mm which was introduced by environmental noise. Equivalent damping ratios of the joints were extracted from the measured damping ratios under different excitation magnitudes. The detail extraction procedure will be introduced in Section 5. 5. Results and discussion To verify the efficiency of the proposed finite element method in predicting the modal parameters of the composite materials, the 1storder damping ratios and resonant frequencies of the cross-ply composites (i.e., C100 and C130), obtained from both the experiment and the simulation were compared as shown in Table 2. From Fig. 6(b), it can be found that due to the influence of the damping effect of the air, the damping ratios of the composite material displayed a linear dependence on the maximum displacement response upon being subjected to the impact forces. The intercept of the curves, which represents the damping ratio of a joint measured from a “zero” excitation magnitude [32], was taken as the equivalent damping ratio of the joints in this study. It was observed that both the resonant frequencies and the damping ratios decreased when the specimen length increased. A comparison of two methods showed that the numerical predictions were found to closely approach the experimental data with a relative error of less than 5%. Fig. 7 shows the damping ratios of the two joints (i.e., J100 and J130) subject to typical torques resulting from the different impact forces. Similarly, the damping ratio showed a linear dependence on the excitation magnitude. Beside the air damping, nonlinear interfacial contact behaviors were also responsible for this phenomenon since the slope of the curves with relatively low torque (1 N∙m) was larger than 21

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Fig. 10. Surface damage caused by excessive torque.

that with high torque (11 N∙m). For both joints, the equivalent damping ratios decreased as the applied torque value increased. Fig. 8 shows the displacement decay curves and frequency spectra of the J100 joint with typical residual torques. It can be observed that the displacement response decays more quickly for the joint with the smaller torque value, while the resonant frequencies increased with an increase in the applied torques. A comparison of the results showed that numerical analysis was found to have a good consistency with the experimental findings. To achieve a quantitative analysis, the graphs of the damping ratios and resonant frequencies of the two joints under increasing torques have been depicted in Fig. 9. In Fig. 9(b) and (d), the frequency shift was calculated from the resonant frequency of the joint with a torque equal to 1 N∙m. It can be concluded from the experimental investigation that for a certain torque range the damping ratios decreased while the resonant frequencies increased when the applied torque for both joints was increased. Taking the J100 joint as a representative result, starting at 1 N∙m, the damping ratios decreased with some fluctuations, until the applied torque reached 13 N∙m. Whereas, the resonant frequencies increased from 1 N∙m until a torque of 11 N∙m was reached, after which the resonant frequencies slightly decreased as the applied torque was increased. This implied that the pressure applied to the specimen, induced by a torque of 11 N∙m, was closed to or exceeded the compressive yield strength of the composite material. Observation on the composite surface using SEM as shown in Fig. 10 indicated fiber breaking after 13 N∙m loading which lead to the decrease of resonance frequency. The damping ratios of the J100 and J130 joints under the fully loose conditions (1 N∙m) increased by 10.3% and 8.5%, respectively, compared to those under the fully tightened conditions (13 N∙m). The resonant frequencies of the joints decreased by 0.59% and 0.91% when subject to the fully loose conditions, compared to those under the fully tightened conditions. A comparison of the results showed that the numerical prediction reached good agreement with the experimental investigation. Relative larger errors occurred in damping ratio prediction compared to that of resonance frequency due to simplified damping ratio was adopted for the contact interfaces (only interfacial tangential sliding was considered and thread friction dissipation was simplified). In order to further study the effect of the contact conditions of rough surfaces on the modal parameters of bolted composite joints, joints with perfectly smooth surfaces were comparatively modelled. The contact properties (i.e., stiffness and energy dissipation) of the perfectly smooth interfaces were independent of the applied pressure due to the unchanging contact area. As a result, the resonant frequencies and damping ratios of the joints with perfect interfaces remained the same, regardless of applied torques. In addition, the results from the

numerical analysis showed that the existence of rough surfaces leads to an increase in the damping ratio (induced by interfacial slip), and a decrease in the resonant frequency (caused by partial contact at the rough interface). From the theoretical analysis of the interfacial properties of rough contact surfaces under different pressures, as shown in Fig. 5, it can be concluded that as the applied torques increase, more asperities will come into elastic contact at the interfaces and asperity sliding will decrease in the tangential direction, which will lead to an increase in the resonant frequency along with a decrease in the damping ratio of the bolted composite joints. 6. Concluding remarks A multiscale numerical model, that considers both microscopic interfacial properties and macroscopic composite properties, has been proposed to simulate the dynamic responses of bolted composite joints. The damping and stiffness of the bolted composite joints were modelled using the complex moduli of both the composites and the rough contact interfaces through the elastic-viscoelastic correspondence principle. The modal parameters of the joints were solved using ABAQUS software through the complex eigenvalue method. From a comparison of the modal parameters between the composite beams (C100 and C130) and the bolted composite joints (J100 and J130) of the same total length, the joints experienced a decrease in the resonant frequency and an increase in the damping ratio. Due to the surface roughness, the contact conditions of the interface of a joint were dependent on the applied torque. As the applied torque value was increased, more contact asperities came into elastic deformation at the interface, which resulted in an increase in the normal and tangential storage moduli of the contact interfaces. While energy dissipation of the interface decreased due to a reduction in asperity sliding in the tangential direction. Consequently, the resonant frequency of the bolted composite joint increased and the damping ratio decreased with increasing applied torques. The results obtained from the numerical predictions and the experimental investigations were found to have good consistency. Due to first-order bending mode was discussed in this study, changes in the interfacial tangential stiffness (much lower than the composite stiffness in the x-direction) caused by different applied torques would not significantly affect the resonance frequency of the joint. On the other side, due to tangential interfacial damping was much higher than that of composite material, therefore damping ratio of the bolted composite joints was more sensitive to the changes in the applied torques. This study has provided a theoretical contact model framework, which has relied on numerical analysis that can be accurately 22

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conducted to facilitate the task of predicting the dynamic responses for bolted composite structures subject to different clamping loads. Our current study has been dedicated to conducting first-order modal analysis of composite joints, and in the future the effect of contact conditions between rough surfaces on the higher-order modal properties of bolted composite structures will be further investigated. High-order resonance frequencies and damping ratios (half-power bandwidth method) will be experimentally measured in the frequency domain. In addition, numerical simulation with regards to modal parameters variation of bolted composite joints subject to vibration fatigue considering interfacial wearing and material damage will be conducted to interpret our previous experiment investigation [3].

[10] [11] [12] [13] [14] [15]

Acknowledgments

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This research was funded by the National Natural Science Foundation of China (Grant No. 11832014) and Hitachi Construction Machinery Co. Ltd. (Japan) as a part of the project HSSF-High strength steel fatigue characteristics (1330-239-0009). The first author would like to acknowledge the Hong Kong Research Grants Council via General Research Funds (No. 15201416 and No. 523313) and the National Natural Science Foundation of China (Grant No. 51635008 and No. 51375414).

[17] [18] [19] [20] [21]

Appendix A. Supplementary data

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Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2018.12.019.

[23] [24]

References

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