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Free vibration analysis of a debonded curved sandwich beam
Accepted Manuscript Free Vibration Analysis of a Debonded Curved Sandwich Beam Ebrahim Sadeghpour, Mojtaba Sadighi, Abdolreza Ohadi PII:
S0997-7538(15)00152-7
DOI:
10.1016/j.euromechsol.2015.11.006
Reference:
EJMSOL 3253
To appear in:
European Journal of Mechanics / A Solids
Received Date: 19 January 2015 Revised Date:
12 November 2015
Accepted Date: 16 November 2015
Please cite this article as: Sadeghpour, E., Sadighi, M., Ohadi, A., Free Vibration Analysis of a Debonded Curved Sandwich Beam, European Journal of Mechanics / A Solids (2015), doi: 10.1016/ j.euromechsol.2015.11.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Free Vibration Analysis of a Debonded Curved Sandwich Beam Ebrahim Sadeghpour1, Mojtaba Sadighi2*, Abdolreza Ohadi3 1
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Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected] 2 Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected] 3 Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected]
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Abstract
In this paper, a high order theory is developed to study the free vibration response
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of a debonded curved sandwich beam. Since the real contact condition at the debonded region is nonlinear, two linear “with contact” and “without contact” models are employed. The Lagrange’s principle and the Rayleigh-Ritz method are employed to derive and solve the governing equations. The model regards the radial and circumferential rigidities of the core. For this purpose, quadratic
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polynomial distributions across the core’s height are proposed for its displacement components. The high order model is validated by finite element simulation in ANSYS Workbench. It will be shown that the quadratic distributions can
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effectively be used for a debonded curved beam comparing to other possible distributions such as a linear pattern which has been previously suggested for an
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undamaged beam. The analyses demonstrate that curvature angle and boundary conditions play important roles in the vibration response of a curved beam. Also, the results show that the debond effect on natural frequencies is approximately identical for flat and curved beams. Keywords: Curved sandwich beam, Debonding, Free Vibration, Sandwich high order theory, Rayleigh-Ritz method
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1. Introduction Sandwich structures are suitable for a broad application in industries owing to their exceptional benefits such as high specific strength and stiffness. A large number of
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studies which have investigated sandwich structures are devoted to flat beams or panels. These structures may be used as curved panels or beams with different curvatures. Therefore, there is a need to develop a theoretical model that could
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evaluate the behavior of curved structures. In addition, a main weakness of sandwiches is their vulnerability to debonding between the stiff facesheets and the
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soft core[1]. As a result, the theoretical model should be applicable for a debonded beam.
Smidt[2] employed the analytical elasticity method to analyze the bending of a curved sandwich beam. Vaswani et al.[3] utilized the energy method to obtain the governing dynamic equation of a curved sandwich beam and also conducted
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vibration tests to verify the model. Their model ignores the effect of the core’s flexibility and assumes that the radial displacement is uniform through the beam’s thickness. Sakiyama et al.[4] applied the Green function to derive the characteristic
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equations of the free vibration of a curved sandwich beam. They neglected the effect of the core’s axial and radial normal stresses and assumed that the shear
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stress is uniform through the core. They stated that a transition may arise between vibration modes when the curvature of the sandwich beam increases. Several refined theories have been proposed to more accurately evaluate displacement, stress and strain components through the thickness of beams or plates. Batra et al.[5] proposed the high order shear and normal deformable plate theory (HOSNDPT) and used Legendre’s polynomial to model the displacements through the thickness. This theory has been employed to investigate the vibration,
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static and dynamic deformations of thick plates and laminates [6-9]. Carrera [10] developed a compact model for multilayered structures to describe different twodimensional theories as a unified formulation. This formulation can address both
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equivalent single layer and layer-wise descriptions and also different zigzag methods. Carrera and Petrolo[11] have studied the influence of high order terms used in refined beam theories. They have reported the necessary order of polynomials is different in various problems to obtain accurate results. Williams
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[12] presented a new unified theory which can deal with general nonlinear and multiscale problems. His model is composed of a superposition of local and
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general displacement components. Demasi [13] proposed a generalized unified model and introduced several new models based on different combinations of layer-wise, zigzag and HSDT models. He applied these combinations to a flat sandwich panel and reported that using a layer-wise or zigzag type of expansion
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results in a better approximation of transverse deformations. Frostig et al. [14] proposed the high order sandwich panel theory (HSAPT) to consider the transverse flexibility of the core. They have supposed that the axial rigidity of the core is negligible and found a distribution for the core’s
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displacement components when the structure is under static loading. Li and Kardomateas[15] tried to improve the accuracy of stress results by employing
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polynomials of fourth and fifth orders for the core’s transverse and in-plane displacement, respectively. Although they have increased the order of polynomial to enhance the accuracy, their model is inaccurate in evaluation of transverse stress in the core. Frostig[16] used the same assumption of negligible circumferential stress to study the bending of curved sandwich beams. He proposed that the core’s shear stress has inverse relation with the square of radial coordinate and derived a new pattern for the core’s displacement components in curved beams.
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Bozhevolnaya and Frostig[17] applied the assumption of HSAPT to study the free vibration response of curved beams. However, they used the pattern, which was proposed by Frostig [13], only for the core’s displacement components and
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assumed that its velocity and acceleration components are linear polynomials of the radial coordinate. Bozhevolnaya and Sun[18] used the linear distributions for the core’s displacement components as well as the velocity and acceleration components. They studied the effects of the boundary conditions and curvature
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angle and reported that the curvature angle has a key role in the vibration behavior
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of a beam with fixed ends.
Several researchers have experimentally or theoretically studied the response of a debonded flat sandwich beam [19-22]. Schwartz-Givli et al.[19, 20, 23] investigated the influence of debonding between core and face on the free vibration and dynamic behavior of sandwich beam using high order theory. They stated that
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the real contact condition in a debonded area is completely nonlinear phenomenon that can be linearized by “with contact” and “without contact” models. These linear contact models have been extensively employed to study the free vibration of debonded composite beams [24-26]. Comparison between experimental and
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theoretical results for composite beams revealed that the “with contact” model compares well with experimental data for different debond length, but the “without
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contact” model predicts lower frequencies as the debond length increases[22, 25, 26]. A few studies have investigated the behavior of a debonded curved beam. Frostig and Thomsen analyzed the bending behavior of a debonded curved beam[27] and also considered the effect of thermal loadings[28]. They used the same higher order model was developed by Frostig [16]. Baba et al. [29] conducted experimental tests on debonded sandwich beams with different curvature angles and compared their test results with finite element simulations. They have reported
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that the existence of a debonding equals to 20% of the beam length has reduced the frequencies of flat and curved beams up to 3% and 10%, respectively, and concluded that a curved beam is more sensitive to debonding. However, the
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difference between their empirical and FEM results is about 10%, which makes their judgment unreliable.
Authors’ literature survey shows that a little attention has been devoted to study
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the dynamic behavior of a curved sandwich beam, especially if a debonding exists between the core and facesheets. In the present study, a higher order theory is
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developed to model the free vibration response of a debonded curved sandwich beam. Although the linear pattern for the core’s displacement components is suitable for the free vibration analysis of an undamaged curved beam, it will be shown that it does not provide accurate results for a debonded beam. Therefore, a quadratic pattern is proposed for the core’s displacement, velocity and acceleration
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components. The facesheets are modeled as the Euler-Bernoulli beams likewise what proposed by Frostig[16]. The results are validated by ANSYS Workbench simulation and presented for different curvature angles, boundary conditions,
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debond lengths and debond positions.
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2. Analytical modeling
A debonded curved sandwich beam of arc-length L and width b is illustrated in Fig.1. The beam consists of a core of height c and two facesheets of thickness dt and db. The subscripts t, b and c stand for the top and bottom facesheets and the core, respectively. The kinematic relations of the facesheets are supposed to follow the Euler-Bernoulli curved beam theory,
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w j (r, φ ) = w 0 j (r, φ ) u j (r, φ ) = u 0 j (φ ) + z j β j r0 j
(1) ( j = t ,b )
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βj =
u 0 j (φ ) −w j (φ ),φ
where r and ϕ denote the radial and circumferential coordinates; r0j is the radius of each facesheet’s centerline; zj is the radial distance from each centerline; u0j and w0j
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are the circumferential and radial deformations of each centerline.
Fig.1. Schematic presentation of a debonded curved beam
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Hence, the circumferential strains of the facesheets read, òφφ j = ò0 j (φ ) + z j κ j (φ ) ò0 j (φ ) =
κ j (φ ) =
u 0 j (φ ),φ + w j (φ ) r0 j
u 0 j (φ ),φ − w j (φ ),φφ r0 j 2
(2) ( j = t ,b )
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where ϵ0j and κj are the normal strain and curvature of the centerlines. Also, the facesheets is considered linear and isotropic,
σ φφ j = E f òφφ j
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(3)
where Ef is the Young’s modulus of the facesheets and σϕϕj is their circumferential normal stress. The radial and circumferential displacements of the core are
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assumed quadratic polynomials of the radial distance from the core’s centerline, w c = w 0c + w 1c ( r − r0 ) + w 2c ( r − r0 ) 2
(4)
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u c = u 0c + u 1c ( r − r0 ) + u 2c ( r − r0 ) 2
where r0 is the radius of the core’s centerline. Six parameters have been expressed in the above equations to describe the displacement fields through the core. u 0c and w 0c are supposed as independent variables. By applying four deformation
compatibility conditions at the core/facesheets interfaces, the four remaining
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variables are stated in terms of u 0c , w 0c and the circumferential and radial displacements of the facesheets,
w 0t − w 0b c 2(w 0b + w 0t − 2w 0c ) w 2c = c2 dt d 1 (u 0t −w 0t ,φ ) − u 0b − b (u 0b − w 0b ,φ )) u 1c = (u 0t − 2rt 2rb c dt d 2 u 2c = 2 (u 0t + (u 0t − w 0t ,φ ) + u 0b + b (u 0b − w 0b ,φ ) − 2u 0c ) c 2rt 2rb
(5)
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w 1c =
where the subscript ,ϕ denotes the circumferential derivative; and rt and rb are the the radii of the top and bottom core/ facesheet interfaces, respectively. Considering small deformation in cylinderical coordinate, the kinematic relations of the core read,
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∂u c w c + r ∂φ r ∂w c òrrc = ∂r ∂u u ∂w c γ r φc = ( c − c + ) ∂r r r ∂φ òφφ c =
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(6)
where ϵϕϕc, ϵrrc and γrϕc are the normal and shear stress components of the core. The present approach regards the circumferential rigidity of the core in contrast to the
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assumption made by Frostig[16]. The core is assumed as a 2-D isotropic body
the strains and displacements,
∂u c w c ∂w c + +ν ) r ∂φ r ∂r ∂w ∂u w = E c (òrrc + ν òφφc ) = E c ( c + ν ( c + c )) ∂r r ∂φ r ∂u u ∂w c = G c γ rφc = G c ( c − c + ) ∂r r r ∂φ
σ φφc = E c (òφφc + ν òrrc ) = E c ( σ rrc
(7)
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τ r φc
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under plane stress condition; hence the stress components can be stated in terms of
where Gc and ν are the shear modulus and the Poisson’s ration of the core.
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2.1. Modeling the debonded region using linear models In a debonded region, two layers can slip in tangential direction which implies on
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nonmatching tangential displacements and null shear stresses on these layers. Besides, the normal contact may exist in the debonded region and the radial displacements are compatible on the debonded layers. This case is called “with contact” condition in contrast to “without contact” condition in which the layers separate each other and the normal stress is null on them. The real contact condition in a debonded region is a combination of the “with contact” and “without contact” conditions which causes nonlinearity in the structure’s behavior. To
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overcome this nonlinearity, the free vibration response of the debonded beam can be studied using the linear “with contact” and “without contact” models [19, 27]. The location of the debonded region is assumed at the top interface; therefore,
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the displacement components are compatible at the bottom interface. In “with contact” model, the circumferential displacement compatibility condition at the top interface is replaced by null shear stress condition, ∂u c u c ∂w c − + =0 r r ∂φ ∂r
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τ r φc = 0 →
(8)
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Hence, the terms describe the circumferential displacement of the core are rewritten,
db 1 1 ∂w c + 2 (u 0b − w 0b ,x )) 2 2 2rb (rb − r0 ) (rt − r0 ) ∂φ r = r 1 1 rt 2 − r0 rb − u 0c [ + ]} (rb − r0 ) 2 (rt 2 − r02 ) (rt 2 − r02 )(rb − r0 ) d 1 ∂w c rb r0 u 2c = {(u 0b + b (u 0b − w 0b ,x )) + − u 0c } 2rb (rb − r0 ) r0∂φ r = r r0 (rb − r0 ) r0 rb − rt 2
u 1c = {(u 0b +
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t
(9)
t
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The radial displacement compatibility condition exists in the “with contact” model and the radial displacement of the core has the same form given in equation
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(5). However, this condition is substituted by null radial normal stress in the “without contact” model, σ rrc = 0
(10)
In order to obtain a simple form for the core’s radial displacement, the core’s radial normal strain is assumed zero at the debonded interface. The terms describe the radial displacement of the core read,
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2rt − r0 − rb 2(rt − r0 )(rb − r0 )3 1 = (w b − w 0c ) (rb − r0 )((rb − r0 ) − 2(rt − r0 ))
w 1c = (w b − w 0c )
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w 2c
(11)
2.2. Solution procedure
The motion equations are derived using the Lagrange’s principle,
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d ∂T ∂V ( )+ =0 dt ∂q& ∂q
(12)
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where T and V are the kinetic and potential energies; and q is the generalized coordinate. It is assumed that the core’s velocity and acceleration components are quadratic polynoimals of radial coordinate similar to what have been stated for displacement components in equation (4). The potential and kinetic energies are
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written in terms of the stress and strain components and displacement fields, 1 V = [ ∫ (σ φφt òφφt )dv + ∫ (σ φφb òφφb )dv bottom 2 top +∫
core
top
ρt (u& t2 + w& t2 )dv + ∫
core
ρc (u& c2 + w& c2 )dv ]
∫
bottom
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+∫
(13)
ρb (u& b2 + w& b2 )dv
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1 T = [ 2
(σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dv ]
The kinetic and potential energies are integrated over fully bonded and debonded regions,
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rb rt b α1 rt +dt { [ r (σ φφt òφφt )dr + ∫ r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]fb d φ ∫ ∫ rb −d b rb 2 0 rt
+∫
α2
α1
+∫
rt +d t
[∫r
rt +d t
[ α ∫r
T =
b 2
2
t
{∫ [∫
α1
rt +d t
0
rt
α2
rt +d t
+∫
α1
+∫
α
t
rt +d t
[∫r
t
r (σ φφt òφφt )dr + ∫
rb
rt
r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]deb d φ rb
rt
rb −d b
ρt r (u&t2 +w& t2 )dr + ∫
rb
ρt r (u&t2 +w& t2)dr + ∫
rb
r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]fb d φ}
rb −d b
rb −db
ρt r (u&t2 +w& t2)dr + ∫
rb
rb −d b
rb
rt
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]fb d φ rb
rt
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]deb d φ rb
rt
(14)
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]fb d φ} rb
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α2
[∫r
rb
rb −d b
t
α
r (σ φφt òφφt )dr + ∫
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V =
where α is the beam’s curvature angle; and the debonded region is located between
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two angles of α1 and α2. The Rayleigh-Ritz method is applied to descritize the motion equations. The response of the beam is assumed to be harmonic with frequency ω. Trigonometric functions are used as trial functions such that satisfy geometry boundary conditions. For instance, the following series can be used for a
direction,
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simply-supported curved beam whose boundaries can move in circumferential
(15)
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u 0t (φ , t ) Qu 0tm cos(m πφ / α ) u 0b (φ , t ) Qu 0bm cos(m πφ / α ) n Qw w 0t (φ , t ) 0tm sin( m πφ / α ) i ωt =e ∑ m =1 Qw 0bm sin( m πφ / α ) w 0b (φ , t ) w 0c (φ , t ) Qw 0cm sin(m πφ / α ) u 0c (φ , t ) Qu 0cm cos(m πφ / α )
The series are truncated by n terms and its value was determined to be 50 after convergence analysis. One may rewrite the above equation to model a beam with circumferentially fixed ends. In this case, cosinus functions are replaced by sinus ones. The trial functions are substituted in the potential and kinetic energies and
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the Lagrange’s principle is applied to obtain the governing free vibration equations, (−ω 2 M + K )Q = 0
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(16)
where ω and Q represent the natural frequency and mode shape vector. K and M are the stiffness and inertia matrices which are obtained by replacing the potential
problem is solved using the Lancoz algorithm.
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2.3. Real contact model
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and kinetic energies into the Lagrange’s principle. The preceding eigen-value
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The real contact condition in a debonded region can be modeled for a problem under static and dynamic loadings. One may model the real contact condition in a dynamic problem to calculate its dynamic response, and then apply the fast Fourier transfer (FFT) method to evaluate natural frequencies. For this purpose, the problem is solved under transient dynamic loadings or specified initial conditions in several time-steps. The debonded region is divided to several zones and a contact function is defined for every zone as follow, (17)
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0 without contact Contact (i ) = i = 1, 2,..., Nd 1 with contact
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The contact condition is checked in every time-step. If contact exists in each timestep, the transverse displacements of the core and debonded facesheet are compatible at their interface: with contact condition :w t = w c (at the int erface )
(19)
Therefore, the radial stress should be examined to determine the contact condition for the next time-step: σ rr < 0 → with contact condition σ rr > 0 → without contact condition
(20)
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On the other hand, if a zone is in “without contact” condition, the radial stress is zero at the interface: without contact condition :σ rr = 0(at the int erface )
(21)
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In this case, the transverse displacements should be examined at the debonded interface: (22)
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w t < w c → without contact condition w t > w c → with contact condition
In this way, the dynamic problem can be solved in every time-step using the
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Newmark numerical integration scheme. After the problem was solved, the FFT method is applied to its response to obtain natural frequencies.
3. Results and Discussion
A curved sandwich beam with the mechanical and geometrical properties given in
parametric studies.
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Table 1 has been considered to verify the developed model and carry out
Table 1. Mechanical and geometrical properties of a curved sandwich beam.
Core
Young’ modolus
Et= Eb =36 GPa
Ec=0.05GPa
Poisson’s ratio
νt= νb =0.3
νc=0.25
Density
ρt =ρb=1800(Kg/m3)
ρc=50(Kg/m3)
Thickness
db=dt=1.5mm
c=20mm
Length
L=300mm
L=300mm
Width
b=20mm
b=20mm
Curvature angle
α=30 deg
α=30 deg
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Facesheets
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3.1. Validation In this study, quadratic polynomial distributions across the core’s height are proposed for the core’s displacement components. The authors have tried several
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other possible distributions to study the free vibration response. The analyses were performed for both intact and debonded curved beams for a wide verity of circumstances such as various curvature angles, boundary conditions, debond
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lengths and debond positions and were compared to ANSYS simulations. The results indicated that the quadratic distribution agrees very well with the ANSYS
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simulation comparing to other distributions. Linear distribution that has been proposed by Bozhevolnaya and Sun [8] is one of the other examined patterns. In this section, the linear and quadratic patterns are termed as u1w1 and u2w2, respectively, and their results are compared for three different cases. A two dimensional model was created in modal environment of ANSYS
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Workbench under plane stress condition. The beam was meshed with quad element as shown in Fig.2. The meshing has been refined around the debonded region to obtain a mesh-size independent solution. ANSYS Workbench uses element plane
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183 by default for linear analyses. The Workbench provides several connection types to model a debonded region. “No separation” type has been used for the
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linear “with contact” model. Also, the connection has been suppressed for the case of “without contact” model.
Fig.2 the meshing of a debonded curved beam in ANSYS Workbench
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Example1) a debonded simply-supported curved beam with the properties given in Table 1 has been considered. It is assumed that the debonded region is located at the midspan of the top interface with a length equals to 13% of the total arc-length.
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The boundaries are allowed to have movement in circumferential direction. Table 2 lists the frequencies calculated by the u1w1 and u2w2 patterns and their differences relative to ANSYS results. Table 2 shows that the results calculated by the u2w2 pattern compare very well with those of the ANSYS simulation either for
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the “with contact” model or “without contact”. However, the difference between
mode.
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the u1w1 pattern and the ANSYS results is high and reaches 15% in the second
Table 2. Natural frequencies of a debonded curved beam with circumferentially movable ends obtained by the high order models and ANSYS. Freq (Hz)
With contact
without contact
ANSYS u2w2 %Diff u1w1
%Diff
ANSYS u2w2
%Diff u1w1
%Diff
362
370
+2.2%
371
+2.4%
362
369
+1.9%
370
+2.2%
f2
772
764
-1.0%
665
-13.9%
772
760
-1.6%
655
-15.2%
f3
1296
1326 +2.3% 1351
+4.3%
1280
1312 +2.5% 1344
+5.0%
f4
1661
1688 +1.6% 1619
-2.5%
1660
1679 +1.1% 1607
-3.2%
f5
2692
2739 +1.7% 2750
+2.2%
2615
2505
2551
-2.4%
f6
3315
3338 +0.7% 3510
+6.0%
2686
2713 +1.0% 2726
+1.5%
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f1
-4.2%
Example2) the boundary conditions of the previous example have been changed to be fixed against circumferential movements and the frequencies are provided in Table 3. Similarly, the u2w2 pattern is in a good agreement with ANSYS in contrast to the u1w1 pattern whose differences to the results of ANSYS reach 16%.
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Table 3. Natural frequencies of a debonded curved beam with circumferentially immovable ends obtained by the high order models and ANSYS.
%Diff
ANSYS u2w2
%Diff u1w1
%Diff
795
750
673
-15.3%
790
745
-5.7%
662
-16.2%
f2
996
1033 +3.7% 1040
+4.4%
995
1032 +3.7% 1040
+4.5%
f3
1444
1431
-0.9%
1449
+0.3%
1443
1417
-1.8%
1441
-0.1%
f4
1657
1643
-0.8%
1584
-4.4%
1656
1629
-1.6%
1569
-5.2%
f5
2286
2303 +0.7% 2397
+4.9%
2086
2075
-0.5%
2138
+2.5%
f6
2710
2716 +0.2% 2742
+1.2%
2650
2526
-4.6%
2574
-2.9%
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f1
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-5.6%
without contact
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With contact Freq (Hz) ANSYS u2w2 %Diff u1w1
Example3) the theoretical models are compared to the ANSYS results in Table 4 for a beam with stiffer core. The beam was illustrated in example1 has been considered and the Young’s modulus of its core has been increased up to five
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times. According to Table 4, the difference between the u1w1 pattern and the ANSYS finite element results reaches 17%.
f1
With contact
ANSYS u2w2 %Diff u1w1
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Freq (Hz)
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Table 4. Natural frequencies of a debonded curved beam with stiffer core and circumferentially movable ends obtained by the high order models and ANSYS.
without contact %Diff
ANSYS u2w2
%Diff u1w1
%Diff +1.5%
587
586
-0.1%
586
-0.1%
577
585
+1.4%
586
1516
1502
-0.9%
1302 -14.1%
1515
1492
-1.5%
1255 -17.2%
2649
2728 +3.0% 2772
+4.6%
2492
2530 +1.5% 2640
+4.3%
f4
3420
3480 +1.8% 3266
-4.5%
3170
3174 +0.1% 3182
+0.4%
f5
4641
4751 +2.8% 4948
+7.4%
3403
3424 +0.1% 3299
-3.1%
f6
5421
5579 +2.9% 5566
+2.7%
4918
5009 +1.8% 5171
+5.1%
f2 f3
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Example4) A beam similar to example1 was considered and the length of the debonded region was increased to 20% of the beam’s arc-length. The middle point of the debonded region (αm, refer to Fig.1) was moved to angular position of α/3.
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Also, the debonded region was divided into a “with contact” and a “without contact” zones with equal lengths. Table 5 compares the results of ANSYS and the u1w1 and u2w2 models. According to Table 5, the differences between the results of the u1w1 and ANSYS models have considerable increased. This example
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demonstrates that the u1w1 pattern is inappropriate in analysis of debonded curved sandwich beams. Also, the mode shapes obtained by the u2w2 model and ANSYS
similar mode shapes.
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are compared in Fig.3. As shown in Fig.3, the u2w2 and ANSYS models predict
Table 5. Natural frequencies of a debonded curved beam with combined “with contact” and “without contact” conditions obtained by the high order models and ANSYS.
u2w2
%Diff
u1w1
%Diff
345
355
+2.9%
332
-3.7%
780
807
+3.3%
759
-2.7%
1128
1181
+4.7%
1094
-3%
1685
1693
+0.7%
1277
-24.2%
f5
2150
2187
+2.3%
1833
-14.7%
f6
2561
2657
+3.7%
2296
-10.3%
f1 f2 f3
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f4
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Freq(Hz) ANSYS
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Fig 3. Comparison of modes shapes of a debonded beam whose debonded region is located at
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angular position α/3 and with partial contact condition.
The above analyses were repeated for beams with larger debonded area and different curvature angles and it was observed that the u2w2 pattern is in a good
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agreement with the ANSYS finite element simulation and the results of the u1w1 model is associated with large errors. The linear distribution employed in the u1w1
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model is inaccurate to calculate the core’s stress components. Since the displacement compatibility conditions are replaced by stress conditions in a debonded beam, this model is not capable of modeling beams with debonded region. Consequently, the u2w2 pattern is used for the next analyses of curved beams and conducting parametric studies.
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3.1.1 Evaluation of stress and strain components The main aim of this paper is to study debonding effects on stiffness and vibration response of a sandwich beam. In a debonded sandwich beam, it is also
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desired to evaluate shear and normal stress fields near the debonded region. For this purpose, values of strain components are presented to assess the accuracy of the proposed high order model in evaluation of stress component for future studies.
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A curved sandwich beam with the properties listed in Table 1 has been considered. The debonded region’s length is 20% of the beam’s length and is located at the
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Fig. 4 for the first three modes.
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midspan. The normalized shear strain of the core at the top interface is plotted in
Fig.4 Normalized shear strain distribution for the first three modes
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According to Fig. 4, the shear strain distribution across the beam is almost identical in both models at points far from the debonded region. However, the difference between two models is relatively high close to the debonded region. Furthermore, the analyses showed that distributions of radial normal strain obtained by the high order model do not match to those of FE modeling. Therefore, the proposed high order model is not suitable to evaluate core’s stress and strain values around a debonded region.
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3.2. Comparing the “with contact” and “without contact” model The simplified linear “with contact” and “without contact” models are used in the free vibration analysis of a debonded beam. The debonded region can be divided
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into several zones with different contact conditions. Therefore, one may suppose infinite contact conditions at the debonded region which are composed of “with contact” and “without contact” zones.
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In order to investigate the effect of contact condition, a debonded region of length 80 mm (26.6% of the beam length) has been considered which is placed at
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the middle of the beam. The debonded region is assumed to consist of two “without contact” zones and a “without contact” zone between them. The “without contact” zone is located at the middle of the debonded region and its length is varied from zero (completely “with contact” case) to the total debond length (completely “without contact” case). Fig.5 presents the variation of the natural
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frequencies with the “without contact” length. In short “without contact” lengths, the frequencies are slightly reduced by increase in the “without contact” length and are similar to those of completely “with contact” condition. In some modes,
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however, the frequencies may sharply decrease when the “without contact” length reaches a critical point. For instance, the frequencies of the two first modes are
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approximately independent of the contact condition. However, the frequency of the third mode significantly decreases in large “without contact” lengths. The decrease in this frequency reaches 36% for the completely “without contact” condition. Furthermore, the slope of reduction is relatively small for the frequency of the fourth mode (the second asymmetric mode), in contrast to the fifth mode (the third symmetric mode). In this way, the order of these modes is changed at a “without contact” length of 45mm.
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Fig.5 The effect of “without contact” length on natural frequencies, the “without contact” zone is located between two “with contact” zones.
In order to examine the effects of contact conditions on mode shapes, the
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facesheets’ radial deformations are presented in Fig.6 for the first six modes and four different “without contact” lengths. The solid and dashed lines in Fig.6 stand for the top and bottom facesheets’ deformations, respectively. The results which
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are shown in the leftmost and rightmost columns of Fig.6 (cases 1 and 4) are related to the completely “with contact” and “without contact” conditions. In cases
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2 and 3, the length of the “without contact” zone is considered to be 50% and 75% of the total debond length. According to Fig.6, the results of different cases are very similar for the first two modes. In the third mode, however, the mode shape changes from global deformations in case 1 to local deformations of the top facesheet at the debonded area. In addition, the order of the fourth and fifth modes is different for the two left columns and two right ones.
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Fig.6 Natural frequencies and mode shapes for four different contact conditions.
In a similar way, the frequencies are presented in Fig.7 for the case that a “with
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contact” zone is located between two “without contact” zones. The length of the “with contact” zone has been varied between the completely “without contact” and “with contact” conditions. In this case, however, the frequencies are almost independent of the “with contact” length and are similar to those of the completely “with contact” condition. It should be noticed that the length of each “without contact” zones is smaller than the half length of the debonded region. Therefore, it
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can be concluded that the effect of contact becomes prominent when a large
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continuous zone is considered in the “without contact” condition.
Fig.7 The effect of the “with contact” length on natural frequencies, the “with contact” zone is
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located between two “without contact” zones.
3.3. Comparing the linearized contact models with real contact condition Natural frequencies of a debonded beam can be obtained by applying the fast
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Fourier transform (FFT) to its dynamic response. This method is similar to an experimental measurement of natural frequencies. Since the real contact condition
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is a nonlinear phenomenon, frequencies of a debonded beam depend on ways which a beam is excited. A curved beam with a debonding of 20% beam’s length has been considered. The beam was excited by an initial impulse force acting near the edge of debonding on the top facesheet. The frequencies have been obtained by taking FFT from transverse displacements of two points at the midspan and the edge of the debonded region on the top facesheet. Fig.8 shows the FFT results and the peak values present natural frequencies.
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Fig 8. FFT diagram of the response of a debonded curved beam
The value of natural frequencies obtained by the linear “with contact” “without contact” models are listed in Table 6. As it is shown in Table 6, the “without contact” model predicts an additional frequency between the third and fourth
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modes which is not evaluated by the two other contact models. Also, the frequencies obtained by the linear “with contact” mode are close to those were obtained by the real contact condition. Consequently, the “with contact” model can
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be used as a simplified model instead of the real contact condition. Table 6. Comparison the results obtained by FFT of the response evaluated by real contact
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condition with those of “with contact” model
“with contact” model
f1(Hz)
FFT of the real contact model 380
370
“without contact” model 367
f2(Hz)
710
729
720
f3(Hz)
1210
1290
1145
-
-
-
1483
f4(Hz)
1630
1684
1667
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3.4. The effect of boundary conditions and curvature angle Analyses show that the effect of curvature angle is related to the boundary conditions. Therefore, natural frequencies of a curved beam will be compared in
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two cases in which the boundaries are free and fixed to move in the circumferential direction. 3.4.1. Undamaged beam
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A beam with the properties given in Table 1 has been considered and its boundaries are allowed to move in the circumferential direction. The frequencies
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have been calculated for different curvature angles, while the beam’s length was kept constant. The dependency of the natural frequencies on the curvature angle is demonstrated in Fig. 9. As shown in Fig. 9, the frequencies slightly decrease with the curvature angle. Also, the symmetric and asymmetric modes of the curved
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beam appear alternately similar to a flat beam.
Fig.9. The effect of curvature angle on natural frequencies of a curved beam with circumferentially free ends
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In the same way, the natural frequencies are provided for the beam whose ends are fixed against circumferential displacement, see Fig.9. In this case, however, Fig. 9-a indicates that the frequencies of the first and second modes do not follow a
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similar pattern. In small curvature angles, a sharp increase is observed in the solid line which corresponds to the first frequency with symmetric mode shape. However, the second mode’s frequency (dashed line) with asymmetric mode shape slowly decreases with the curvature angle. The dashed and solid lines intersect at a
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curvature angle of 25deg. This angle is named here as a mode switching point in which the first and second modes have identical frequencies. Beyond this point, the
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first symmetric mode gets higher frequency and becomes the second mode of the beam.
Similarly, the variations of natural frequencies for the higher modes are plotted in Fig. 10-b and 10-c. The same scenario could be observed for the higher modes
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except that the mode switching occurs in greater curvature angles. For instance, the mode switching points between the third and fourth modes and also the fifth and
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sixth modes are at angles of 55 and 78 deg, respectively.
Fig.10 The effect of curvature angle on natural frequencies of a curved beam with circumferentially fixed ends: a) the first and second modes, b) the third and fourth modes, c) the fifth and sixth modes
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The preceding analyses show that the free vibration response of a curved sandwich beam is considerably affected by the circumferential movement of its boundaries. Bozhevolnaya and Sun[18] reported this result by comparison of
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simply-supported and clamped curved beams. They justified it as the coupling between the circumferential and flexural motions. To have a deeper insight into this problem, the circumferential and radial displacements of the top facesheet are provided in Fig.11 for the first four modes of a beam with circumferentially free
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ends and curvature angle of 30deg. According to Fig.11, the vibration amplitude in the circumferential direction (dashed line) is very small comparing to the radial
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direction (solid line), except for the first mode, which is about one fifth of the radial displacement. Consequently, when the movement is constrained in the circumferential direction, the beam gets stiffer in this mode and the corresponding
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frequency increases.
Fig.11 The mode shapes of a curved beam with circumferential free ends and curvature angle of 30deg, solid line: radial displacement, dashed line: circumferential displacement.
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3.4.2. Debonded beam In this section, the effects of curvature angle and boundary conditions are analyzed for a debonded beam. In the other words, it would be answered if the effect of
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debonding is more prominent for a curved beam or a flat one. The frequencies have been calculated using the “with contact” model for different curvature angles and normalized by what were obtained for the undamaged beam. The beam is the same
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as specified in Table 1 and the debond length is 20% of the beam’s length.
At first, the normalized frequencies are presented for the beam with
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circumferentially movable ends, see Fig.12. The results show that the effect of
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debonding is not accentuated by increase in curvature angle.
Fig.12 The effect of curvature angle on a debonded beam with circumferential free ends, the frequencies have been normalized by those of the intact beam.
The analyses have been repeated for the beam with circumferentially fixed ends. In this case, the changes in frequencies are plotted in Fig.13 for the first six modes.
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Fig.13 illustrates that the existence of debonding alters the location of the switching mode point. It is because that the debonding does not identically affect
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different modes as it will be discussed in the next sections.
Fig.13. The effect of curvature angle on natural frequencies of a debonded curved beam with circumferentially fixed ends: a) the first and second modes, b) the third and fourth modes, c) the
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fifth and sixth modes
Also, the frequencies have been normalized by those were obtained for the undamaged beam, see Fig.14. The frequencies should be normalized by the corresponding mode that could be noticed by the help of the mode shapes. As
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shown in Fig.14-a, the increase in curvature angle has reduced the effect of debonding for the first asymmetric mode, while it has increased this effect for the
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first symmetric mode. The analyses for higher modes also show that the effect of debonding may slightly be increased, decreased or remain unchanged by the variation of curvature angle. Baba et al. [29] reported that the reductions in natural frequencies due to debonding are greater for a curved beam. In this way, it can be noticed that their conclusion is not general or is accompanied with errors.
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Fig.14. The effect of curvature angle on a debonded beam with circumferential fixed ends, the
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frequencies have been normalized by those of the intact beam.
3.5. The effects of debond length and position
The effect of debond length is studied by applying the “with contact” model. A beam with the properties given in Table 1 has been considered. It has been
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assumed that the debonded region is located at the midspan of the top interface and its length has been increased up to one third of the beam’s length. The frequencies have been normalized by those of the undamaged beam. The results for the first three modes are plotted in Fig.14. The calculations have been performed for both
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cases in which the beam’s boundaries can move in circumferential direction (solid line) and are fixed in this direction (dashed line). Fig. 14 shows that a debonded
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area of 33% the beam’s length can reduce the frequencies up to 20%.
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Fig.15. Effect of debond length on the first three frequencies of a debonded beam obtained by the
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“with contact” model. solid line: the beam with circumferential free ends, dashed line: the beam with circumferential fixed ends.
The debond position is the other important factor which controls the significance
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of a debonding. A debonded region of 13% total beam length has been considered. The middle point of the debonded region, αm (refer to Fig.1), has been changed
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along the beam span and the variations of frequencies are presented in Fig.16-(a).
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Fig.16. a) Effect of debond location on the first three natural frequencies, obtained by “with contact” model b) shear stress distribution for the intact beam corresponding to the first three
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natural frequencies.
According to Fig.16-a, the maximum values of the graphs are very close to frequencies of a similar intact beam. The first three frequencies for the intact beam
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are 371, 857 and 1336 Hz, respectively, which means that the debonded region may be located where it has a negligible effect on a particular mode. On the other
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hand, the reduction of frequencies in the first three modes may be raised up to 13.5%, 11% and 9.6%, respectively. To give an explanation for the changes in the frequencies, the core’s shear stress distribution at the top interface is depicted in Fig.16-b for the intact beam. Comparing the graphs of Fig.16-a and 16-b for each mode shows that if the debonded region is located at a position with low shear stress, it has a slight influence on that mode. As the debonded region is located closer to a position with high shear stress, it more reduces the frequency of that mode.
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3.4. Conclusion A high order sandwich theory was developed to evaluate the free vibration behavior of a debonded curved beam. The following points can be summarized
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based on the analysis was performed,
- The debonded region may be divided into “with contact” and “without contact” zones. The contact effect becomes prominent when a large “without
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contact” zone is encountered. The increase in the length of a continuous “without contact” zone, which is not interrupted by “with contact” zones,
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may considerably reduce the natural frequencies. Also, it changes the mode shapes to have local deformation at the debonded region rather than global deformation.
- A real contact condition model has been developed to evaluate dynamic response of a debonded beam. Natural frequencies can be obtained by
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applying the FFT to the beam’s dynamic response evaluated by the real contact model. Comparing frequencies obtained by the real contact condition with linear model justifies using the “with contact” model.
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- In a curved beam with circumferentially free ends, increase in curvature angle slowly decreases the natural frequencies. When the boundaries are
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fixed against movements in this direction, however, the frequencies of the symmetric modes remarkably increase with curvature angle, while the asymmetric modes slowly decrease. In this way, the sequences of modes are changed at specific angles, namely the mode switching points.
- In beams with free ends, the effect of debonding is identical for beams with different curvature angles. When the boundaries are fixed against the circumferential movements, a change in curvature angle may slightly increase or decrease the effect of debonding.
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- The debonding has minor effect in small debonded region. As it is expected the increase in debond length decreases the frequencies. For instance, a debonded region of one third of the beam length can reduce the frequencies
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up to 20%. - The debonding does not influence all modes in a similar way and it depends on the position of the debonded region. If the debonded area is located
prominent effect on that mode.
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3.5. References
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where the core’s shear stress is high in a certain mode, it has a more
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1. Carlsson LA, Kardomateas GA. Structural and failure mechanics of sandwich composites. Solid mechanics and its applications. 2011, Dordrecht ; New York: Springer. 2. Smidt S. Bending of curved sandwich beams. Composite Structures, 1995. 33: 211-225. 3.Vaswani J, Asnani NT, Nakra BC. Vibration and Damping Analysis of Curved Sandwich Beams with a Viscoelastic Core. Composite Structures, 1988; 10:231245.
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4. Sakiyama T, Matsuda H, Morita C. Free vibration analysis of sandwich arches with elastic or viscoelastic core and various kinds of axis-shape and boundary conditions. Journal of sound and vibration, 1997; 203:505-522.
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5. Batra, R, Vidoli, S, Vestroni, F. Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories. Journal of Sound and vibration, 2002; 257:63-88. 6. Batra, R, Aimmanee, S. Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories. Computers & Structures, 2005; 83:934-955. 7. Qian, L, Batra, R, Chen, L. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov – Galerkin method. Composites: Part B, 2004; 35: 685–697. 8. Qian, L, Batra,R, Chen, L. Elastostatic Deformations of a Thick Plate by using a Higher-Order Shear and Normal Deformable Plate Theory and two Meshless Local
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Petrov-Galerkin (MLPG) Methods. Computer Modeling in Engineering and Sciences, 2003; 4: 161-175. 9. Xiao, JR, Gilhooley, DF, Batra, RC, Gillespie, JW, McCarthy, MA , Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method. Composite Part B, 2008; 39: 414– 427.
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10. Carrera, E., Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering, 2003; 10:215-296. 11. Carrera, E, Petrolo, M. On the Effectiveness of Higher-Order Terms in Refined Beam Theories. Journal of Applied Mechanics, 2011; 78:1-17. 12. Williams, T.O. A new theoretical framework for the formulation of general, nonlinear, multiscale plate theories. International Journal of Solids and Structures, 2008; 45:2534-2560. 13. Demasi, L. Partially layer wise advanced zig zag and HSDT models based on the generalized unified formulation. Engineering Structures, 2013; 53:63-91. 14. Frostig Y, Baruch M, Vilnay O, Sheinman I. High-order theory for sandwichbeam behavior with transversely flexible core. Journal of Engeeniring Mechanics 1992; 118:1026-1043.
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15. Li, R, Kardomateas, GA. Nonlinear high-order core theory for sandwich plates with orthotropic phases. AIAA journal, 2008; 46:2926-2934. 16. Frostig Y. Bending of Curved Sandwich Panels with a Transversely Flexible Core-Closed-Form High-Order Theory. Journal of Sandwich Structures and Materials, 1999; 1:4-41. 17. Bozhevolnaya E, Frostig Y. Free Vibrations of Curved Sandwich Beams with a Transversely Flexible Core. Journal of Sandwich structures and materials, 2001; 3:311-342. 18. Bozhevolnaya E, Sun JQ. Free Vibration Analysis of Curved Sandwich Beams. Journal of Sandwich structures and materials, 2004; 6:47-73. 19. Schwarts-Givli H, Rabinovitch O, FrostigY. Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core - a modified Galerkin approach. Journal of sound and vibration, 2007; 301:253–277. 20. Schwarts-Givli H, Rabinovitch O, Frostig Y. Free Vibration of Delaminated Unidirectional Sandwich Panels with a Transversely Flexible Core and General Boundary Conditions – A High-Order Approach. Journal of Sandwich structures and materials 2008; 10:99-131.
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21. Baba B, Thoppul S. Experimental evaluation of the vibration behavior of flat and curved sandwich composite beams with face/core debond. Composite Structures, 2009; 91:110–119
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22. Shu D. Vibration of sandwich beams with double delaminations. Composites science and technology, 1995. 54(1): p. 101-109. 23. Schwarts-Givli H, Rabinovitch O, Frostig Y. Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core—a modified Galerkin approach. Journal of sound and vibration, 2007. 301:253–277. 24. Lee J. Free vibration analysis of delaminated composite beams. Computers & Structures, 2000. 74(2): p. 121-129. 25. Luo H, Hanagud S. Dynamics of delaminated beams. International Journal of Solids and Structures, 2000. 37(10): 1501-1519. 26. Shen MH, Grady J. Free vibrations of delaminated beams. AIAA journal, 1992. 30(5): 1361-1370. 27. Frostig Y, Thomsen T, Vinson RJ. High-order Bending Analysis of Unidirectional Curved ‘‘Soft’’ Sandwich Panels with Disbonds and Slipping Layers. Journal of Sandwich structures and materials, 2004; 6:167-194. 28. Frostig Y, Thomsen T. Non-linear thermo-mechanical behaviour of delaminated curved sandwich panels with a compliant core. International Journal of Solids and Structures, 2011. 48:2218-2237. 29. Baba B, Thoppul S, Gibson RF. Experimental and Numerical Investigation of Free Vibrations of Composite Sandwich Beams with Curvature and Debonds. Experimental Mechanics, 2011. 51:857-868.
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Highlights • Free vibration response of a debonded curved sandwich beam is studied
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• A high order theory is developed for curved sandwich beams
• Debonding is modeled using “With contact” and “without contact” models
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• The curvature angle strongly affects the vibration response of curved beams
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• The effect of curvature angle is in conjunction with boundary conditions
S0997-7538(15)00152-7
DOI:
10.1016/j.euromechsol.2015.11.006
Reference:
EJMSOL 3253
To appear in:
European Journal of Mechanics / A Solids
Received Date: 19 January 2015 Revised Date:
12 November 2015
Accepted Date: 16 November 2015
Please cite this article as: Sadeghpour, E., Sadighi, M., Ohadi, A., Free Vibration Analysis of a Debonded Curved Sandwich Beam, European Journal of Mechanics / A Solids (2015), doi: 10.1016/ j.euromechsol.2015.11.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Free Vibration Analysis of a Debonded Curved Sandwich Beam Ebrahim Sadeghpour1, Mojtaba Sadighi2*, Abdolreza Ohadi3 1
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Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected] 2 Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected] 3 Department of Mechanical Engineering, Amirkabir University of Technology, Iran; [email protected]
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Abstract
In this paper, a high order theory is developed to study the free vibration response
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of a debonded curved sandwich beam. Since the real contact condition at the debonded region is nonlinear, two linear “with contact” and “without contact” models are employed. The Lagrange’s principle and the Rayleigh-Ritz method are employed to derive and solve the governing equations. The model regards the radial and circumferential rigidities of the core. For this purpose, quadratic
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polynomial distributions across the core’s height are proposed for its displacement components. The high order model is validated by finite element simulation in ANSYS Workbench. It will be shown that the quadratic distributions can
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effectively be used for a debonded curved beam comparing to other possible distributions such as a linear pattern which has been previously suggested for an
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undamaged beam. The analyses demonstrate that curvature angle and boundary conditions play important roles in the vibration response of a curved beam. Also, the results show that the debond effect on natural frequencies is approximately identical for flat and curved beams. Keywords: Curved sandwich beam, Debonding, Free Vibration, Sandwich high order theory, Rayleigh-Ritz method
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1. Introduction Sandwich structures are suitable for a broad application in industries owing to their exceptional benefits such as high specific strength and stiffness. A large number of
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studies which have investigated sandwich structures are devoted to flat beams or panels. These structures may be used as curved panels or beams with different curvatures. Therefore, there is a need to develop a theoretical model that could
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evaluate the behavior of curved structures. In addition, a main weakness of sandwiches is their vulnerability to debonding between the stiff facesheets and the
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soft core[1]. As a result, the theoretical model should be applicable for a debonded beam.
Smidt[2] employed the analytical elasticity method to analyze the bending of a curved sandwich beam. Vaswani et al.[3] utilized the energy method to obtain the governing dynamic equation of a curved sandwich beam and also conducted
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vibration tests to verify the model. Their model ignores the effect of the core’s flexibility and assumes that the radial displacement is uniform through the beam’s thickness. Sakiyama et al.[4] applied the Green function to derive the characteristic
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equations of the free vibration of a curved sandwich beam. They neglected the effect of the core’s axial and radial normal stresses and assumed that the shear
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stress is uniform through the core. They stated that a transition may arise between vibration modes when the curvature of the sandwich beam increases. Several refined theories have been proposed to more accurately evaluate displacement, stress and strain components through the thickness of beams or plates. Batra et al.[5] proposed the high order shear and normal deformable plate theory (HOSNDPT) and used Legendre’s polynomial to model the displacements through the thickness. This theory has been employed to investigate the vibration,
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static and dynamic deformations of thick plates and laminates [6-9]. Carrera [10] developed a compact model for multilayered structures to describe different twodimensional theories as a unified formulation. This formulation can address both
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equivalent single layer and layer-wise descriptions and also different zigzag methods. Carrera and Petrolo[11] have studied the influence of high order terms used in refined beam theories. They have reported the necessary order of polynomials is different in various problems to obtain accurate results. Williams
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[12] presented a new unified theory which can deal with general nonlinear and multiscale problems. His model is composed of a superposition of local and
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general displacement components. Demasi [13] proposed a generalized unified model and introduced several new models based on different combinations of layer-wise, zigzag and HSDT models. He applied these combinations to a flat sandwich panel and reported that using a layer-wise or zigzag type of expansion
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results in a better approximation of transverse deformations. Frostig et al. [14] proposed the high order sandwich panel theory (HSAPT) to consider the transverse flexibility of the core. They have supposed that the axial rigidity of the core is negligible and found a distribution for the core’s
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displacement components when the structure is under static loading. Li and Kardomateas[15] tried to improve the accuracy of stress results by employing
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polynomials of fourth and fifth orders for the core’s transverse and in-plane displacement, respectively. Although they have increased the order of polynomial to enhance the accuracy, their model is inaccurate in evaluation of transverse stress in the core. Frostig[16] used the same assumption of negligible circumferential stress to study the bending of curved sandwich beams. He proposed that the core’s shear stress has inverse relation with the square of radial coordinate and derived a new pattern for the core’s displacement components in curved beams.
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Bozhevolnaya and Frostig[17] applied the assumption of HSAPT to study the free vibration response of curved beams. However, they used the pattern, which was proposed by Frostig [13], only for the core’s displacement components and
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assumed that its velocity and acceleration components are linear polynomials of the radial coordinate. Bozhevolnaya and Sun[18] used the linear distributions for the core’s displacement components as well as the velocity and acceleration components. They studied the effects of the boundary conditions and curvature
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angle and reported that the curvature angle has a key role in the vibration behavior
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of a beam with fixed ends.
Several researchers have experimentally or theoretically studied the response of a debonded flat sandwich beam [19-22]. Schwartz-Givli et al.[19, 20, 23] investigated the influence of debonding between core and face on the free vibration and dynamic behavior of sandwich beam using high order theory. They stated that
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the real contact condition in a debonded area is completely nonlinear phenomenon that can be linearized by “with contact” and “without contact” models. These linear contact models have been extensively employed to study the free vibration of debonded composite beams [24-26]. Comparison between experimental and
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theoretical results for composite beams revealed that the “with contact” model compares well with experimental data for different debond length, but the “without
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contact” model predicts lower frequencies as the debond length increases[22, 25, 26]. A few studies have investigated the behavior of a debonded curved beam. Frostig and Thomsen analyzed the bending behavior of a debonded curved beam[27] and also considered the effect of thermal loadings[28]. They used the same higher order model was developed by Frostig [16]. Baba et al. [29] conducted experimental tests on debonded sandwich beams with different curvature angles and compared their test results with finite element simulations. They have reported
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that the existence of a debonding equals to 20% of the beam length has reduced the frequencies of flat and curved beams up to 3% and 10%, respectively, and concluded that a curved beam is more sensitive to debonding. However, the
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difference between their empirical and FEM results is about 10%, which makes their judgment unreliable.
Authors’ literature survey shows that a little attention has been devoted to study
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the dynamic behavior of a curved sandwich beam, especially if a debonding exists between the core and facesheets. In the present study, a higher order theory is
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developed to model the free vibration response of a debonded curved sandwich beam. Although the linear pattern for the core’s displacement components is suitable for the free vibration analysis of an undamaged curved beam, it will be shown that it does not provide accurate results for a debonded beam. Therefore, a quadratic pattern is proposed for the core’s displacement, velocity and acceleration
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components. The facesheets are modeled as the Euler-Bernoulli beams likewise what proposed by Frostig[16]. The results are validated by ANSYS Workbench simulation and presented for different curvature angles, boundary conditions,
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debond lengths and debond positions.
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2. Analytical modeling
A debonded curved sandwich beam of arc-length L and width b is illustrated in Fig.1. The beam consists of a core of height c and two facesheets of thickness dt and db. The subscripts t, b and c stand for the top and bottom facesheets and the core, respectively. The kinematic relations of the facesheets are supposed to follow the Euler-Bernoulli curved beam theory,
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w j (r, φ ) = w 0 j (r, φ ) u j (r, φ ) = u 0 j (φ ) + z j β j r0 j
(1) ( j = t ,b )
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βj =
u 0 j (φ ) −w j (φ ),φ
where r and ϕ denote the radial and circumferential coordinates; r0j is the radius of each facesheet’s centerline; zj is the radial distance from each centerline; u0j and w0j
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are the circumferential and radial deformations of each centerline.
Fig.1. Schematic presentation of a debonded curved beam
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Hence, the circumferential strains of the facesheets read, òφφ j = ò0 j (φ ) + z j κ j (φ ) ò0 j (φ ) =
κ j (φ ) =
u 0 j (φ ),φ + w j (φ ) r0 j
u 0 j (φ ),φ − w j (φ ),φφ r0 j 2
(2) ( j = t ,b )
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where ϵ0j and κj are the normal strain and curvature of the centerlines. Also, the facesheets is considered linear and isotropic,
σ φφ j = E f òφφ j
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(3)
where Ef is the Young’s modulus of the facesheets and σϕϕj is their circumferential normal stress. The radial and circumferential displacements of the core are
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assumed quadratic polynomials of the radial distance from the core’s centerline, w c = w 0c + w 1c ( r − r0 ) + w 2c ( r − r0 ) 2
(4)
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u c = u 0c + u 1c ( r − r0 ) + u 2c ( r − r0 ) 2
where r0 is the radius of the core’s centerline. Six parameters have been expressed in the above equations to describe the displacement fields through the core. u 0c and w 0c are supposed as independent variables. By applying four deformation
compatibility conditions at the core/facesheets interfaces, the four remaining
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variables are stated in terms of u 0c , w 0c and the circumferential and radial displacements of the facesheets,
w 0t − w 0b c 2(w 0b + w 0t − 2w 0c ) w 2c = c2 dt d 1 (u 0t −w 0t ,φ ) − u 0b − b (u 0b − w 0b ,φ )) u 1c = (u 0t − 2rt 2rb c dt d 2 u 2c = 2 (u 0t + (u 0t − w 0t ,φ ) + u 0b + b (u 0b − w 0b ,φ ) − 2u 0c ) c 2rt 2rb
(5)
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w 1c =
where the subscript ,ϕ denotes the circumferential derivative; and rt and rb are the the radii of the top and bottom core/ facesheet interfaces, respectively. Considering small deformation in cylinderical coordinate, the kinematic relations of the core read,
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∂u c w c + r ∂φ r ∂w c òrrc = ∂r ∂u u ∂w c γ r φc = ( c − c + ) ∂r r r ∂φ òφφ c =
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(6)
where ϵϕϕc, ϵrrc and γrϕc are the normal and shear stress components of the core. The present approach regards the circumferential rigidity of the core in contrast to the
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assumption made by Frostig[16]. The core is assumed as a 2-D isotropic body
the strains and displacements,
∂u c w c ∂w c + +ν ) r ∂φ r ∂r ∂w ∂u w = E c (òrrc + ν òφφc ) = E c ( c + ν ( c + c )) ∂r r ∂φ r ∂u u ∂w c = G c γ rφc = G c ( c − c + ) ∂r r r ∂φ
σ φφc = E c (òφφc + ν òrrc ) = E c ( σ rrc
(7)
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τ r φc
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under plane stress condition; hence the stress components can be stated in terms of
where Gc and ν are the shear modulus and the Poisson’s ration of the core.
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2.1. Modeling the debonded region using linear models In a debonded region, two layers can slip in tangential direction which implies on
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nonmatching tangential displacements and null shear stresses on these layers. Besides, the normal contact may exist in the debonded region and the radial displacements are compatible on the debonded layers. This case is called “with contact” condition in contrast to “without contact” condition in which the layers separate each other and the normal stress is null on them. The real contact condition in a debonded region is a combination of the “with contact” and “without contact” conditions which causes nonlinearity in the structure’s behavior. To
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overcome this nonlinearity, the free vibration response of the debonded beam can be studied using the linear “with contact” and “without contact” models [19, 27]. The location of the debonded region is assumed at the top interface; therefore,
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the displacement components are compatible at the bottom interface. In “with contact” model, the circumferential displacement compatibility condition at the top interface is replaced by null shear stress condition, ∂u c u c ∂w c − + =0 r r ∂φ ∂r
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τ r φc = 0 →
(8)
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Hence, the terms describe the circumferential displacement of the core are rewritten,
db 1 1 ∂w c + 2 (u 0b − w 0b ,x )) 2 2 2rb (rb − r0 ) (rt − r0 ) ∂φ r = r 1 1 rt 2 − r0 rb − u 0c [ + ]} (rb − r0 ) 2 (rt 2 − r02 ) (rt 2 − r02 )(rb − r0 ) d 1 ∂w c rb r0 u 2c = {(u 0b + b (u 0b − w 0b ,x )) + − u 0c } 2rb (rb − r0 ) r0∂φ r = r r0 (rb − r0 ) r0 rb − rt 2
u 1c = {(u 0b +
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t
(9)
t
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The radial displacement compatibility condition exists in the “with contact” model and the radial displacement of the core has the same form given in equation
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(5). However, this condition is substituted by null radial normal stress in the “without contact” model, σ rrc = 0
(10)
In order to obtain a simple form for the core’s radial displacement, the core’s radial normal strain is assumed zero at the debonded interface. The terms describe the radial displacement of the core read,
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2rt − r0 − rb 2(rt − r0 )(rb − r0 )3 1 = (w b − w 0c ) (rb − r0 )((rb − r0 ) − 2(rt − r0 ))
w 1c = (w b − w 0c )
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w 2c
(11)
2.2. Solution procedure
The motion equations are derived using the Lagrange’s principle,
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d ∂T ∂V ( )+ =0 dt ∂q& ∂q
(12)
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where T and V are the kinetic and potential energies; and q is the generalized coordinate. It is assumed that the core’s velocity and acceleration components are quadratic polynoimals of radial coordinate similar to what have been stated for displacement components in equation (4). The potential and kinetic energies are
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written in terms of the stress and strain components and displacement fields, 1 V = [ ∫ (σ φφt òφφt )dv + ∫ (σ φφb òφφb )dv bottom 2 top +∫
core
top
ρt (u& t2 + w& t2 )dv + ∫
core
ρc (u& c2 + w& c2 )dv ]
∫
bottom
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+∫
(13)
ρb (u& b2 + w& b2 )dv
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1 T = [ 2
(σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dv ]
The kinetic and potential energies are integrated over fully bonded and debonded regions,
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rb rt b α1 rt +dt { [ r (σ φφt òφφt )dr + ∫ r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]fb d φ ∫ ∫ rb −d b rb 2 0 rt
+∫
α2
α1
+∫
rt +d t
[∫r
rt +d t
[ α ∫r
T =
b 2
2
t
{∫ [∫
α1
rt +d t
0
rt
α2
rt +d t
+∫
α1
+∫
α
t
rt +d t
[∫r
t
r (σ φφt òφφt )dr + ∫
rb
rt
r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]deb d φ rb
rt
rb −d b
ρt r (u&t2 +w& t2 )dr + ∫
rb
ρt r (u&t2 +w& t2)dr + ∫
rb
r (σ φφb òφφb )dr + ∫ r (σ φφc òφφc + σ rrc òrrc + τ rφc γ rφc )dr ]fb d φ}
rb −d b
rb −db
ρt r (u&t2 +w& t2)dr + ∫
rb
rb −d b
rb
rt
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]fb d φ rb
rt
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]deb d φ rb
rt
(14)
ρb r (u&b2 +w& b2 )dr + ∫ ρc r (u&c2 +w& c2 )dr ]fb d φ} rb
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α2
[∫r
rb
rb −d b
t
α
r (σ φφt òφφt )dr + ∫
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V =
where α is the beam’s curvature angle; and the debonded region is located between
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two angles of α1 and α2. The Rayleigh-Ritz method is applied to descritize the motion equations. The response of the beam is assumed to be harmonic with frequency ω. Trigonometric functions are used as trial functions such that satisfy geometry boundary conditions. For instance, the following series can be used for a
direction,
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simply-supported curved beam whose boundaries can move in circumferential
(15)
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u 0t (φ , t ) Qu 0tm cos(m πφ / α ) u 0b (φ , t ) Qu 0bm cos(m πφ / α ) n Qw w 0t (φ , t ) 0tm sin( m πφ / α ) i ωt =e ∑ m =1 Qw 0bm sin( m πφ / α ) w 0b (φ , t ) w 0c (φ , t ) Qw 0cm sin(m πφ / α ) u 0c (φ , t ) Qu 0cm cos(m πφ / α )
The series are truncated by n terms and its value was determined to be 50 after convergence analysis. One may rewrite the above equation to model a beam with circumferentially fixed ends. In this case, cosinus functions are replaced by sinus ones. The trial functions are substituted in the potential and kinetic energies and
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the Lagrange’s principle is applied to obtain the governing free vibration equations, (−ω 2 M + K )Q = 0
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(16)
where ω and Q represent the natural frequency and mode shape vector. K and M are the stiffness and inertia matrices which are obtained by replacing the potential
problem is solved using the Lancoz algorithm.
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2.3. Real contact model
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and kinetic energies into the Lagrange’s principle. The preceding eigen-value
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The real contact condition in a debonded region can be modeled for a problem under static and dynamic loadings. One may model the real contact condition in a dynamic problem to calculate its dynamic response, and then apply the fast Fourier transfer (FFT) method to evaluate natural frequencies. For this purpose, the problem is solved under transient dynamic loadings or specified initial conditions in several time-steps. The debonded region is divided to several zones and a contact function is defined for every zone as follow, (17)
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0 without contact Contact (i ) = i = 1, 2,..., Nd 1 with contact
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The contact condition is checked in every time-step. If contact exists in each timestep, the transverse displacements of the core and debonded facesheet are compatible at their interface: with contact condition :w t = w c (at the int erface )
(19)
Therefore, the radial stress should be examined to determine the contact condition for the next time-step: σ rr < 0 → with contact condition σ rr > 0 → without contact condition
(20)
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On the other hand, if a zone is in “without contact” condition, the radial stress is zero at the interface: without contact condition :σ rr = 0(at the int erface )
(21)
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In this case, the transverse displacements should be examined at the debonded interface: (22)
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w t < w c → without contact condition w t > w c → with contact condition
In this way, the dynamic problem can be solved in every time-step using the
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Newmark numerical integration scheme. After the problem was solved, the FFT method is applied to its response to obtain natural frequencies.
3. Results and Discussion
A curved sandwich beam with the mechanical and geometrical properties given in
parametric studies.
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Table 1 has been considered to verify the developed model and carry out
Table 1. Mechanical and geometrical properties of a curved sandwich beam.
Core
Young’ modolus
Et= Eb =36 GPa
Ec=0.05GPa
Poisson’s ratio
νt= νb =0.3
νc=0.25
Density
ρt =ρb=1800(Kg/m3)
ρc=50(Kg/m3)
Thickness
db=dt=1.5mm
c=20mm
Length
L=300mm
L=300mm
Width
b=20mm
b=20mm
Curvature angle
α=30 deg
α=30 deg
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Facesheets
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3.1. Validation In this study, quadratic polynomial distributions across the core’s height are proposed for the core’s displacement components. The authors have tried several
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other possible distributions to study the free vibration response. The analyses were performed for both intact and debonded curved beams for a wide verity of circumstances such as various curvature angles, boundary conditions, debond
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lengths and debond positions and were compared to ANSYS simulations. The results indicated that the quadratic distribution agrees very well with the ANSYS
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simulation comparing to other distributions. Linear distribution that has been proposed by Bozhevolnaya and Sun [8] is one of the other examined patterns. In this section, the linear and quadratic patterns are termed as u1w1 and u2w2, respectively, and their results are compared for three different cases. A two dimensional model was created in modal environment of ANSYS
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Workbench under plane stress condition. The beam was meshed with quad element as shown in Fig.2. The meshing has been refined around the debonded region to obtain a mesh-size independent solution. ANSYS Workbench uses element plane
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183 by default for linear analyses. The Workbench provides several connection types to model a debonded region. “No separation” type has been used for the
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linear “with contact” model. Also, the connection has been suppressed for the case of “without contact” model.
Fig.2 the meshing of a debonded curved beam in ANSYS Workbench
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Example1) a debonded simply-supported curved beam with the properties given in Table 1 has been considered. It is assumed that the debonded region is located at the midspan of the top interface with a length equals to 13% of the total arc-length.
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The boundaries are allowed to have movement in circumferential direction. Table 2 lists the frequencies calculated by the u1w1 and u2w2 patterns and their differences relative to ANSYS results. Table 2 shows that the results calculated by the u2w2 pattern compare very well with those of the ANSYS simulation either for
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the “with contact” model or “without contact”. However, the difference between
mode.
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the u1w1 pattern and the ANSYS results is high and reaches 15% in the second
Table 2. Natural frequencies of a debonded curved beam with circumferentially movable ends obtained by the high order models and ANSYS. Freq (Hz)
With contact
without contact
ANSYS u2w2 %Diff u1w1
%Diff
ANSYS u2w2
%Diff u1w1
%Diff
362
370
+2.2%
371
+2.4%
362
369
+1.9%
370
+2.2%
f2
772
764
-1.0%
665
-13.9%
772
760
-1.6%
655
-15.2%
f3
1296
1326 +2.3% 1351
+4.3%
1280
1312 +2.5% 1344
+5.0%
f4
1661
1688 +1.6% 1619
-2.5%
1660
1679 +1.1% 1607
-3.2%
f5
2692
2739 +1.7% 2750
+2.2%
2615
2505
2551
-2.4%
f6
3315
3338 +0.7% 3510
+6.0%
2686
2713 +1.0% 2726
+1.5%
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f1
-4.2%
Example2) the boundary conditions of the previous example have been changed to be fixed against circumferential movements and the frequencies are provided in Table 3. Similarly, the u2w2 pattern is in a good agreement with ANSYS in contrast to the u1w1 pattern whose differences to the results of ANSYS reach 16%.
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Table 3. Natural frequencies of a debonded curved beam with circumferentially immovable ends obtained by the high order models and ANSYS.
%Diff
ANSYS u2w2
%Diff u1w1
%Diff
795
750
673
-15.3%
790
745
-5.7%
662
-16.2%
f2
996
1033 +3.7% 1040
+4.4%
995
1032 +3.7% 1040
+4.5%
f3
1444
1431
-0.9%
1449
+0.3%
1443
1417
-1.8%
1441
-0.1%
f4
1657
1643
-0.8%
1584
-4.4%
1656
1629
-1.6%
1569
-5.2%
f5
2286
2303 +0.7% 2397
+4.9%
2086
2075
-0.5%
2138
+2.5%
f6
2710
2716 +0.2% 2742
+1.2%
2650
2526
-4.6%
2574
-2.9%
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f1
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-5.6%
without contact
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With contact Freq (Hz) ANSYS u2w2 %Diff u1w1
Example3) the theoretical models are compared to the ANSYS results in Table 4 for a beam with stiffer core. The beam was illustrated in example1 has been considered and the Young’s modulus of its core has been increased up to five
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times. According to Table 4, the difference between the u1w1 pattern and the ANSYS finite element results reaches 17%.
f1
With contact
ANSYS u2w2 %Diff u1w1
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Freq (Hz)
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Table 4. Natural frequencies of a debonded curved beam with stiffer core and circumferentially movable ends obtained by the high order models and ANSYS.
without contact %Diff
ANSYS u2w2
%Diff u1w1
%Diff +1.5%
587
586
-0.1%
586
-0.1%
577
585
+1.4%
586
1516
1502
-0.9%
1302 -14.1%
1515
1492
-1.5%
1255 -17.2%
2649
2728 +3.0% 2772
+4.6%
2492
2530 +1.5% 2640
+4.3%
f4
3420
3480 +1.8% 3266
-4.5%
3170
3174 +0.1% 3182
+0.4%
f5
4641
4751 +2.8% 4948
+7.4%
3403
3424 +0.1% 3299
-3.1%
f6
5421
5579 +2.9% 5566
+2.7%
4918
5009 +1.8% 5171
+5.1%
f2 f3
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Example4) A beam similar to example1 was considered and the length of the debonded region was increased to 20% of the beam’s arc-length. The middle point of the debonded region (αm, refer to Fig.1) was moved to angular position of α/3.
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Also, the debonded region was divided into a “with contact” and a “without contact” zones with equal lengths. Table 5 compares the results of ANSYS and the u1w1 and u2w2 models. According to Table 5, the differences between the results of the u1w1 and ANSYS models have considerable increased. This example
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demonstrates that the u1w1 pattern is inappropriate in analysis of debonded curved sandwich beams. Also, the mode shapes obtained by the u2w2 model and ANSYS
similar mode shapes.
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are compared in Fig.3. As shown in Fig.3, the u2w2 and ANSYS models predict
Table 5. Natural frequencies of a debonded curved beam with combined “with contact” and “without contact” conditions obtained by the high order models and ANSYS.
u2w2
%Diff
u1w1
%Diff
345
355
+2.9%
332
-3.7%
780
807
+3.3%
759
-2.7%
1128
1181
+4.7%
1094
-3%
1685
1693
+0.7%
1277
-24.2%
f5
2150
2187
+2.3%
1833
-14.7%
f6
2561
2657
+3.7%
2296
-10.3%
f1 f2 f3
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f4
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Freq(Hz) ANSYS
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Fig 3. Comparison of modes shapes of a debonded beam whose debonded region is located at
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angular position α/3 and with partial contact condition.
The above analyses were repeated for beams with larger debonded area and different curvature angles and it was observed that the u2w2 pattern is in a good
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agreement with the ANSYS finite element simulation and the results of the u1w1 model is associated with large errors. The linear distribution employed in the u1w1
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model is inaccurate to calculate the core’s stress components. Since the displacement compatibility conditions are replaced by stress conditions in a debonded beam, this model is not capable of modeling beams with debonded region. Consequently, the u2w2 pattern is used for the next analyses of curved beams and conducting parametric studies.
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3.1.1 Evaluation of stress and strain components The main aim of this paper is to study debonding effects on stiffness and vibration response of a sandwich beam. In a debonded sandwich beam, it is also
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desired to evaluate shear and normal stress fields near the debonded region. For this purpose, values of strain components are presented to assess the accuracy of the proposed high order model in evaluation of stress component for future studies.
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A curved sandwich beam with the properties listed in Table 1 has been considered. The debonded region’s length is 20% of the beam’s length and is located at the
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Fig. 4 for the first three modes.
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midspan. The normalized shear strain of the core at the top interface is plotted in
Fig.4 Normalized shear strain distribution for the first three modes
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According to Fig. 4, the shear strain distribution across the beam is almost identical in both models at points far from the debonded region. However, the difference between two models is relatively high close to the debonded region. Furthermore, the analyses showed that distributions of radial normal strain obtained by the high order model do not match to those of FE modeling. Therefore, the proposed high order model is not suitable to evaluate core’s stress and strain values around a debonded region.
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3.2. Comparing the “with contact” and “without contact” model The simplified linear “with contact” and “without contact” models are used in the free vibration analysis of a debonded beam. The debonded region can be divided
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into several zones with different contact conditions. Therefore, one may suppose infinite contact conditions at the debonded region which are composed of “with contact” and “without contact” zones.
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In order to investigate the effect of contact condition, a debonded region of length 80 mm (26.6% of the beam length) has been considered which is placed at
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the middle of the beam. The debonded region is assumed to consist of two “without contact” zones and a “without contact” zone between them. The “without contact” zone is located at the middle of the debonded region and its length is varied from zero (completely “with contact” case) to the total debond length (completely “without contact” case). Fig.5 presents the variation of the natural
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frequencies with the “without contact” length. In short “without contact” lengths, the frequencies are slightly reduced by increase in the “without contact” length and are similar to those of completely “with contact” condition. In some modes,
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however, the frequencies may sharply decrease when the “without contact” length reaches a critical point. For instance, the frequencies of the two first modes are
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approximately independent of the contact condition. However, the frequency of the third mode significantly decreases in large “without contact” lengths. The decrease in this frequency reaches 36% for the completely “without contact” condition. Furthermore, the slope of reduction is relatively small for the frequency of the fourth mode (the second asymmetric mode), in contrast to the fifth mode (the third symmetric mode). In this way, the order of these modes is changed at a “without contact” length of 45mm.
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Fig.5 The effect of “without contact” length on natural frequencies, the “without contact” zone is located between two “with contact” zones.
In order to examine the effects of contact conditions on mode shapes, the
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facesheets’ radial deformations are presented in Fig.6 for the first six modes and four different “without contact” lengths. The solid and dashed lines in Fig.6 stand for the top and bottom facesheets’ deformations, respectively. The results which
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are shown in the leftmost and rightmost columns of Fig.6 (cases 1 and 4) are related to the completely “with contact” and “without contact” conditions. In cases
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2 and 3, the length of the “without contact” zone is considered to be 50% and 75% of the total debond length. According to Fig.6, the results of different cases are very similar for the first two modes. In the third mode, however, the mode shape changes from global deformations in case 1 to local deformations of the top facesheet at the debonded area. In addition, the order of the fourth and fifth modes is different for the two left columns and two right ones.
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Fig.6 Natural frequencies and mode shapes for four different contact conditions.
In a similar way, the frequencies are presented in Fig.7 for the case that a “with
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contact” zone is located between two “without contact” zones. The length of the “with contact” zone has been varied between the completely “without contact” and “with contact” conditions. In this case, however, the frequencies are almost independent of the “with contact” length and are similar to those of the completely “with contact” condition. It should be noticed that the length of each “without contact” zones is smaller than the half length of the debonded region. Therefore, it
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can be concluded that the effect of contact becomes prominent when a large
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continuous zone is considered in the “without contact” condition.
Fig.7 The effect of the “with contact” length on natural frequencies, the “with contact” zone is
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located between two “without contact” zones.
3.3. Comparing the linearized contact models with real contact condition Natural frequencies of a debonded beam can be obtained by applying the fast
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Fourier transform (FFT) to its dynamic response. This method is similar to an experimental measurement of natural frequencies. Since the real contact condition
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is a nonlinear phenomenon, frequencies of a debonded beam depend on ways which a beam is excited. A curved beam with a debonding of 20% beam’s length has been considered. The beam was excited by an initial impulse force acting near the edge of debonding on the top facesheet. The frequencies have been obtained by taking FFT from transverse displacements of two points at the midspan and the edge of the debonded region on the top facesheet. Fig.8 shows the FFT results and the peak values present natural frequencies.
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Fig 8. FFT diagram of the response of a debonded curved beam
The value of natural frequencies obtained by the linear “with contact” “without contact” models are listed in Table 6. As it is shown in Table 6, the “without contact” model predicts an additional frequency between the third and fourth
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modes which is not evaluated by the two other contact models. Also, the frequencies obtained by the linear “with contact” mode are close to those were obtained by the real contact condition. Consequently, the “with contact” model can
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be used as a simplified model instead of the real contact condition. Table 6. Comparison the results obtained by FFT of the response evaluated by real contact
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condition with those of “with contact” model
“with contact” model
f1(Hz)
FFT of the real contact model 380
370
“without contact” model 367
f2(Hz)
710
729
720
f3(Hz)
1210
1290
1145
-
-
-
1483
f4(Hz)
1630
1684
1667
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3.4. The effect of boundary conditions and curvature angle Analyses show that the effect of curvature angle is related to the boundary conditions. Therefore, natural frequencies of a curved beam will be compared in
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two cases in which the boundaries are free and fixed to move in the circumferential direction. 3.4.1. Undamaged beam
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A beam with the properties given in Table 1 has been considered and its boundaries are allowed to move in the circumferential direction. The frequencies
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have been calculated for different curvature angles, while the beam’s length was kept constant. The dependency of the natural frequencies on the curvature angle is demonstrated in Fig. 9. As shown in Fig. 9, the frequencies slightly decrease with the curvature angle. Also, the symmetric and asymmetric modes of the curved
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beam appear alternately similar to a flat beam.
Fig.9. The effect of curvature angle on natural frequencies of a curved beam with circumferentially free ends
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In the same way, the natural frequencies are provided for the beam whose ends are fixed against circumferential displacement, see Fig.9. In this case, however, Fig. 9-a indicates that the frequencies of the first and second modes do not follow a
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similar pattern. In small curvature angles, a sharp increase is observed in the solid line which corresponds to the first frequency with symmetric mode shape. However, the second mode’s frequency (dashed line) with asymmetric mode shape slowly decreases with the curvature angle. The dashed and solid lines intersect at a
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curvature angle of 25deg. This angle is named here as a mode switching point in which the first and second modes have identical frequencies. Beyond this point, the
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first symmetric mode gets higher frequency and becomes the second mode of the beam.
Similarly, the variations of natural frequencies for the higher modes are plotted in Fig. 10-b and 10-c. The same scenario could be observed for the higher modes
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except that the mode switching occurs in greater curvature angles. For instance, the mode switching points between the third and fourth modes and also the fifth and
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sixth modes are at angles of 55 and 78 deg, respectively.
Fig.10 The effect of curvature angle on natural frequencies of a curved beam with circumferentially fixed ends: a) the first and second modes, b) the third and fourth modes, c) the fifth and sixth modes
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The preceding analyses show that the free vibration response of a curved sandwich beam is considerably affected by the circumferential movement of its boundaries. Bozhevolnaya and Sun[18] reported this result by comparison of
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simply-supported and clamped curved beams. They justified it as the coupling between the circumferential and flexural motions. To have a deeper insight into this problem, the circumferential and radial displacements of the top facesheet are provided in Fig.11 for the first four modes of a beam with circumferentially free
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ends and curvature angle of 30deg. According to Fig.11, the vibration amplitude in the circumferential direction (dashed line) is very small comparing to the radial
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direction (solid line), except for the first mode, which is about one fifth of the radial displacement. Consequently, when the movement is constrained in the circumferential direction, the beam gets stiffer in this mode and the corresponding
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frequency increases.
Fig.11 The mode shapes of a curved beam with circumferential free ends and curvature angle of 30deg, solid line: radial displacement, dashed line: circumferential displacement.
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3.4.2. Debonded beam In this section, the effects of curvature angle and boundary conditions are analyzed for a debonded beam. In the other words, it would be answered if the effect of
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debonding is more prominent for a curved beam or a flat one. The frequencies have been calculated using the “with contact” model for different curvature angles and normalized by what were obtained for the undamaged beam. The beam is the same
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as specified in Table 1 and the debond length is 20% of the beam’s length.
At first, the normalized frequencies are presented for the beam with
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circumferentially movable ends, see Fig.12. The results show that the effect of
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debonding is not accentuated by increase in curvature angle.
Fig.12 The effect of curvature angle on a debonded beam with circumferential free ends, the frequencies have been normalized by those of the intact beam.
The analyses have been repeated for the beam with circumferentially fixed ends. In this case, the changes in frequencies are plotted in Fig.13 for the first six modes.
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Fig.13 illustrates that the existence of debonding alters the location of the switching mode point. It is because that the debonding does not identically affect
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different modes as it will be discussed in the next sections.
Fig.13. The effect of curvature angle on natural frequencies of a debonded curved beam with circumferentially fixed ends: a) the first and second modes, b) the third and fourth modes, c) the
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fifth and sixth modes
Also, the frequencies have been normalized by those were obtained for the undamaged beam, see Fig.14. The frequencies should be normalized by the corresponding mode that could be noticed by the help of the mode shapes. As
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shown in Fig.14-a, the increase in curvature angle has reduced the effect of debonding for the first asymmetric mode, while it has increased this effect for the
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first symmetric mode. The analyses for higher modes also show that the effect of debonding may slightly be increased, decreased or remain unchanged by the variation of curvature angle. Baba et al. [29] reported that the reductions in natural frequencies due to debonding are greater for a curved beam. In this way, it can be noticed that their conclusion is not general or is accompanied with errors.
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Fig.14. The effect of curvature angle on a debonded beam with circumferential fixed ends, the
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frequencies have been normalized by those of the intact beam.
3.5. The effects of debond length and position
The effect of debond length is studied by applying the “with contact” model. A beam with the properties given in Table 1 has been considered. It has been
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assumed that the debonded region is located at the midspan of the top interface and its length has been increased up to one third of the beam’s length. The frequencies have been normalized by those of the undamaged beam. The results for the first three modes are plotted in Fig.14. The calculations have been performed for both
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cases in which the beam’s boundaries can move in circumferential direction (solid line) and are fixed in this direction (dashed line). Fig. 14 shows that a debonded
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area of 33% the beam’s length can reduce the frequencies up to 20%.
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Fig.15. Effect of debond length on the first three frequencies of a debonded beam obtained by the
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“with contact” model. solid line: the beam with circumferential free ends, dashed line: the beam with circumferential fixed ends.
The debond position is the other important factor which controls the significance
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of a debonding. A debonded region of 13% total beam length has been considered. The middle point of the debonded region, αm (refer to Fig.1), has been changed
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along the beam span and the variations of frequencies are presented in Fig.16-(a).
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Fig.16. a) Effect of debond location on the first three natural frequencies, obtained by “with contact” model b) shear stress distribution for the intact beam corresponding to the first three
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natural frequencies.
According to Fig.16-a, the maximum values of the graphs are very close to frequencies of a similar intact beam. The first three frequencies for the intact beam
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are 371, 857 and 1336 Hz, respectively, which means that the debonded region may be located where it has a negligible effect on a particular mode. On the other
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hand, the reduction of frequencies in the first three modes may be raised up to 13.5%, 11% and 9.6%, respectively. To give an explanation for the changes in the frequencies, the core’s shear stress distribution at the top interface is depicted in Fig.16-b for the intact beam. Comparing the graphs of Fig.16-a and 16-b for each mode shows that if the debonded region is located at a position with low shear stress, it has a slight influence on that mode. As the debonded region is located closer to a position with high shear stress, it more reduces the frequency of that mode.
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3.4. Conclusion A high order sandwich theory was developed to evaluate the free vibration behavior of a debonded curved beam. The following points can be summarized
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based on the analysis was performed,
- The debonded region may be divided into “with contact” and “without contact” zones. The contact effect becomes prominent when a large “without
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contact” zone is encountered. The increase in the length of a continuous “without contact” zone, which is not interrupted by “with contact” zones,
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may considerably reduce the natural frequencies. Also, it changes the mode shapes to have local deformation at the debonded region rather than global deformation.
- A real contact condition model has been developed to evaluate dynamic response of a debonded beam. Natural frequencies can be obtained by
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applying the FFT to the beam’s dynamic response evaluated by the real contact model. Comparing frequencies obtained by the real contact condition with linear model justifies using the “with contact” model.
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- In a curved beam with circumferentially free ends, increase in curvature angle slowly decreases the natural frequencies. When the boundaries are
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fixed against movements in this direction, however, the frequencies of the symmetric modes remarkably increase with curvature angle, while the asymmetric modes slowly decrease. In this way, the sequences of modes are changed at specific angles, namely the mode switching points.
- In beams with free ends, the effect of debonding is identical for beams with different curvature angles. When the boundaries are fixed against the circumferential movements, a change in curvature angle may slightly increase or decrease the effect of debonding.
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- The debonding has minor effect in small debonded region. As it is expected the increase in debond length decreases the frequencies. For instance, a debonded region of one third of the beam length can reduce the frequencies
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up to 20%. - The debonding does not influence all modes in a similar way and it depends on the position of the debonded region. If the debonded area is located
prominent effect on that mode.
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3.5. References
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where the core’s shear stress is high in a certain mode, it has a more
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1. Carlsson LA, Kardomateas GA. Structural and failure mechanics of sandwich composites. Solid mechanics and its applications. 2011, Dordrecht ; New York: Springer. 2. Smidt S. Bending of curved sandwich beams. Composite Structures, 1995. 33: 211-225. 3.Vaswani J, Asnani NT, Nakra BC. Vibration and Damping Analysis of Curved Sandwich Beams with a Viscoelastic Core. Composite Structures, 1988; 10:231245.
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4. Sakiyama T, Matsuda H, Morita C. Free vibration analysis of sandwich arches with elastic or viscoelastic core and various kinds of axis-shape and boundary conditions. Journal of sound and vibration, 1997; 203:505-522.
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5. Batra, R, Vidoli, S, Vestroni, F. Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories. Journal of Sound and vibration, 2002; 257:63-88. 6. Batra, R, Aimmanee, S. Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories. Computers & Structures, 2005; 83:934-955. 7. Qian, L, Batra, R, Chen, L. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov – Galerkin method. Composites: Part B, 2004; 35: 685–697. 8. Qian, L, Batra,R, Chen, L. Elastostatic Deformations of a Thick Plate by using a Higher-Order Shear and Normal Deformable Plate Theory and two Meshless Local
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Petrov-Galerkin (MLPG) Methods. Computer Modeling in Engineering and Sciences, 2003; 4: 161-175. 9. Xiao, JR, Gilhooley, DF, Batra, RC, Gillespie, JW, McCarthy, MA , Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method. Composite Part B, 2008; 39: 414– 427.
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10. Carrera, E., Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering, 2003; 10:215-296. 11. Carrera, E, Petrolo, M. On the Effectiveness of Higher-Order Terms in Refined Beam Theories. Journal of Applied Mechanics, 2011; 78:1-17. 12. Williams, T.O. A new theoretical framework for the formulation of general, nonlinear, multiscale plate theories. International Journal of Solids and Structures, 2008; 45:2534-2560. 13. Demasi, L. Partially layer wise advanced zig zag and HSDT models based on the generalized unified formulation. Engineering Structures, 2013; 53:63-91. 14. Frostig Y, Baruch M, Vilnay O, Sheinman I. High-order theory for sandwichbeam behavior with transversely flexible core. Journal of Engeeniring Mechanics 1992; 118:1026-1043.
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15. Li, R, Kardomateas, GA. Nonlinear high-order core theory for sandwich plates with orthotropic phases. AIAA journal, 2008; 46:2926-2934. 16. Frostig Y. Bending of Curved Sandwich Panels with a Transversely Flexible Core-Closed-Form High-Order Theory. Journal of Sandwich Structures and Materials, 1999; 1:4-41. 17. Bozhevolnaya E, Frostig Y. Free Vibrations of Curved Sandwich Beams with a Transversely Flexible Core. Journal of Sandwich structures and materials, 2001; 3:311-342. 18. Bozhevolnaya E, Sun JQ. Free Vibration Analysis of Curved Sandwich Beams. Journal of Sandwich structures and materials, 2004; 6:47-73. 19. Schwarts-Givli H, Rabinovitch O, FrostigY. Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core - a modified Galerkin approach. Journal of sound and vibration, 2007; 301:253–277. 20. Schwarts-Givli H, Rabinovitch O, Frostig Y. Free Vibration of Delaminated Unidirectional Sandwich Panels with a Transversely Flexible Core and General Boundary Conditions – A High-Order Approach. Journal of Sandwich structures and materials 2008; 10:99-131.
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21. Baba B, Thoppul S. Experimental evaluation of the vibration behavior of flat and curved sandwich composite beams with face/core debond. Composite Structures, 2009; 91:110–119
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22. Shu D. Vibration of sandwich beams with double delaminations. Composites science and technology, 1995. 54(1): p. 101-109. 23. Schwarts-Givli H, Rabinovitch O, Frostig Y. Free vibrations of delaminated unidirectional sandwich panels with a transversely flexible core—a modified Galerkin approach. Journal of sound and vibration, 2007. 301:253–277. 24. Lee J. Free vibration analysis of delaminated composite beams. Computers & Structures, 2000. 74(2): p. 121-129. 25. Luo H, Hanagud S. Dynamics of delaminated beams. International Journal of Solids and Structures, 2000. 37(10): 1501-1519. 26. Shen MH, Grady J. Free vibrations of delaminated beams. AIAA journal, 1992. 30(5): 1361-1370. 27. Frostig Y, Thomsen T, Vinson RJ. High-order Bending Analysis of Unidirectional Curved ‘‘Soft’’ Sandwich Panels with Disbonds and Slipping Layers. Journal of Sandwich structures and materials, 2004; 6:167-194. 28. Frostig Y, Thomsen T. Non-linear thermo-mechanical behaviour of delaminated curved sandwich panels with a compliant core. International Journal of Solids and Structures, 2011. 48:2218-2237. 29. Baba B, Thoppul S, Gibson RF. Experimental and Numerical Investigation of Free Vibrations of Composite Sandwich Beams with Curvature and Debonds. Experimental Mechanics, 2011. 51:857-868.
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Highlights • Free vibration response of a debonded curved sandwich beam is studied
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• A high order theory is developed for curved sandwich beams
• Debonding is modeled using “With contact” and “without contact” models
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• The curvature angle strongly affects the vibration response of curved beams
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• The effect of curvature angle is in conjunction with boundary conditions