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Power reflection and transmission in beam structures containing a semi-infinite crack
Acta Mechanica Solida Sinica, Vol. 21, No. 2, April, 2008 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-008-0821-6
ISSN 0894-9166
POWER REFLECTION AND TRANSMISSION IN BEAM STRUCTURES CONTAINING A SEMI-INFINITE CRACK Li Zhou
Wanchun Yuan
(College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China)
Received 8 December 2006; revision received 8 January 2008
ABSTRACT Wave reflection and transmission in a beam containing a semi-infinite crack are studied analytically based on Timoshenko beam theory. Two kinds of crack surface conditions: non-contact (open) and fully contact (closed) cracks, are considered respectively for an isotropic beam. The analytical solution of reflection and transmission coefficients for a semi-infinite crack is obtained. The power reflection and transmission ratios depend on both the frequency and the position of the crack. Numerical results show the conservation of power transport. The transmitted energy among various wave modes is also investigated. A finite element method is used to verify the validity of the analytical results.
KEY WORDS reflection and transmission, crack, power transport, structural health monitoring
I. INTRODUCTION One of the commonly encountered types of defect or damage in structures is crack. The crack not only causes reduction in stiffness but also affects the strength and integrity of the structure, leading to its final failure. The crack also affects vibration and stability characteristics of beam type structures. Vibrations of cracked beams has been investigated[1] . The vibration characteristics of the split beam, such as natural frequencies and mode shapes, have been examined. As expected, the change of these modal parameters is insensitive to the extent of the damage for the first few natural frequencies. When the transient waves propagate in the beam containing delamination[2, 3] , the damaged region generates reflection and transmission waves which carry information on the nature of the damage. Obviously the transmitted flexural wave velocity in the delaminated region is decreased due to the reduction of overall bending stiffness. None of the studies considered the power transport of transient wave packet, which can be useful for damage identification. Power flow in beam-like structures can be obtained from both theory and experiments[4, 5] . To our best knowledge, no power relationship between the incident wave and the reflected or transmitted waves has been presented. Bazer and Burridge[6] derived a general solution of power flow of the plane wave at an interface in a three-dimensional medium. They were concerned with the energy balance associated with the reflection and refraction of monochromatic plane waves governed by the differential equations at a plane interface or boundary. The power flow in the incident wave is equal to the sum of those in the reflected wave and refraction wave, indicating power conservation. Wang and Rose[7] investigated the wave propagation in beams containing delamination and inhomogeneity. The closed delamination form was only considered, and the results of symmetric Project supported by the National Natural Science Foundation of China (Nos. 50478037 and 10572058) and the Research Foundation for the Doctoral Program of Higher Education (No. 20050287016).
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delamination were given in the paper to compare with the results of inhomogeneity. In this study, power reflected from and transmitted through a crack is proposed to analyze the wave behavior in the split beam-like structure. The purpose of this research is to present an analytical method to determine the power reflected from and transmitted through a semi-infinite crack beam. The beam is modeled using Timoshenko beam theory. To model the cracked region, two extreme cases of crack surface conditions during the wave propagation are considered. One is that the crack surfaces are completely open, which means that there is no contact between two sub-beams. The transmitted waves in sub-beams are independent of each other. The other is that the crack surfaces are completely closed, implying there is a contact pressure between the surfaces. These two cases are shown in Figs.1(a) and 1(b), respectively. The two cases are described as open and closed cracks respectively.
Fig. 1. Transient wave propagation in beams containing a semi-infinite crack.
The sections are divided as follows. Dispersion relations of the open and closed cracks are analyzed in §II. The reflection and transmission coefficients are derived in §III. In §IV the power reflection and transmission are analyzed. Numerical results are provided in §V. The comparison of results from analytical solution and finite element method is made in §VI. Finally conclusions are drawn in §VII.
II. DISPERSION RELATION In the beam model the centroidal axis is taken on the x-axis. Based on Timoshenko beam theory, the general displacements of the beam can be described as U (x, z, t) = u(x, t) + zψ(x, t),
W (x, z, t) = w(x, t)
(1)
where u(x, t) is the axial displacement of the beam in the x-direction; ψ(x, t) is the rotation of the cross section of the beam about the y-axis; w(x, t) is the transverse displacement of the beam in the z-direction. The equilibrium equations for the isotropic beam can be written as N,x = ρA¨ u,
V,x = ρAw, ¨
M,x − V = ρI ψ¨
(2)
where N = χAu,x , V = κ2 GA(w,x + ψ), M = χIψ,x (3) and N , V , and M are the axial force, shear force, and bending moment per unit beam width of the beam; χ = E(1 − μ)/[(1 + μ)(1 − 2μ)], E and G are Young’s modulus and shear modulus respectively; κ2 = π 2 /12 is the transverse shear correction factor; A and I are the cross-sectional area and mass moment of inertia of the beam, and ρ is the mass density. For plane wave solutions the displacements can be represented by u = U0 ei(kx−ωt) ,
w = W0 ei(kx−ωt) ,
ψ = Ψ0 ei(kx−ωt)
There are four roots of flexural wave numbers ⎡ ⎤ 2 2 1/2 2 2 c 1 ω c c 1 l 1 + 2l 2 ± 1 − 2l 2 ⎦ + kj = ⎣ 2 κ cs qω 4 κ cs cl
(4)
(j = 1, 2)
(5)
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and k3 − k1 , k4 = −k2 . In the above, cl = χ/ρ, cs = G/ρ, q = I/A. In a relatively low frequency range, the Timoshenko beam experiences a pair of propagating waves (one positive-going and one negative-going) plus two evanescent (near-field) waves, with k purely imaginary. The pair of waves is called the lowest fundamental flexural A0 wave modes. These two near-field waves can be regarded as positive- and negative-going attenuating waves which decay exponentially. Beyond a cut-off frequency, two pairs of propagating waves (two positive-going and two negative-going) co-exist, called A0 and A1 modes. Thus, with an increase of frequency, a non-propagating mode can decay more slowly and eventually becomes a propagating mode. The cut-off frequency can be simply determined by setting k2 = 0, as κcs (6) ωc = q Phase velocities and group velocities are given by cp =
ω , k
cg =
dω dk
(7)
When frequencies approach infinity, the wave velocity becomes non-dispersive. The group velocities of two flexural modes can be simply derived as cg1 = κcs ,
cg2 = cl
as
ω→∞
(8)
A general solution of Eq.(4) is given by w(x, t) =
4
j=1
aj ei(kj x−ωt) ,
ψ(x, t) =
4
Fj aj ei(kj x−ωt)
(9)
j=1
where Fj = (ω 2 − κ2 c2s kj2 )/(iκ2 c2s kj ) (j = 1, 2, 3, 4) and the amplitudes aj may be complex. For the extensional wave mode which is non-dispersive based on the Timoshenko beam theory, the wave number is given by ω ke = (10) cl
III. REFLECTION AND TRANSMISSION COEFFICIENTS It is well known that a propagating wave is incident upon a discontinuity, which gives rise to wave reflection from and transmission through the discontinuity whose amplitudes and phases are given by reflection and transmission coefficients. The incident wave induces propagating and non-propagating evanescent near-field waves. In this section, a propagating flexural wave at far field excited on a beam incident upon a semi-infinite crack is studied. Two extreme cases of crack surface conditions: non-contact (open) and fully contact (closed) cracks, are considered respectively. The reflection and transmission coefficients of both open and closed cracks are derived separately. 3.1. Open Crack Considering a slender beam containing a semi-infinite crack shown in Fig.1(a), the origin of the coordinate is located at the tip of the crack. The beam can be divided into two regions. The left is the un-cracked region which contains both the incident wave and reflected wave, while the right region is the split beam. Since the crack surfaces are open, there is no contact pressure between the surfaces. Only positive-going wave exists in the two sub-beams, which is referred to as the transmitted wave. In the left region (x ≤ 0), it includes the positive-going incident waves and negative-going reflected waves. It can be written as w0 = aeik1 x + beik2 x + ar e−ik1 x + br e−ik2 x ψ0 = F1 aeik1 x + F2 beik2 x − F1 ar e−ik1 x − F2 br e−ik2 x
(11)
When the crack is not located on the mid-plane, an extensional wave is reflected from and transmitted into two split beam regions. The negative-going reflected extensional wave in the left part can be expressed as u0 = cr e−ike x (12)
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where the time dependence term e−iωt has been suppressed here and hereafter. When the excitation frequency is below the cut-off frequency, i.e., ω < ωc , the second wave mode is evanescent since the wave number k2 is purely imaginary. The sign of k2 is chosen as negative to ensure the finiteness of the amplitude as x → −∞. In the right region (x ≥ 0), the displacement of two sub-beams are denoted by u1 , w1 , ψ1 , u2 , w2 , ψ2 . The subscripts 1 and 2 denote the upper and lower sub-beams respectively. As there are only transmitted waves in this region, the general solution of displacement can be written as (n)
(n)
wn = at eik1 x + bt eik2 x (n) (n) (n) (n) (n) (n) ψn = F1 at eik1 x + F2 bt eik2 x (n) (n) un = ct eike x (n)
(n)
(13)
where index n = 1, 2, denoting the parameters associated with each sub-beam. These parameters can be readily obtained by substituting the stiffness and moment of inertia pertaining to the sub-beams in Eqs.(3). The wave number of the extensional wave is independent of beam thickness according to (1) (2) Eqs.(13), implying ke = ke = ke . (1)
(1)
(1)
(2)
(2)
(2)
are The amplitudes of these waves involving nine unknowns ar , br , cr , at , bt , ct , at , bt , ct given by reflection and transmission coefficients which can be determined from the following continuity and equilibrium conditions at the crack tip, x = 0. h2 h1 ψ0 = ψ1 = ψ2 , u1 = u0 + ψ0 , u2 = u0 − ψ0 2 2 (14) h2 h1 N0 = N1 + N2 , M0 = M1 + M2 + N1 − N2 , V0 = V1 + V2 2 2 Substituting Eqs.(11)-(13) together with Eqs.(3) into Eqs.(14), the equilibrium equations can be written in the matrix form, ⎡ ⎤⎧ a ⎫ ⎡ ⎤ r ⎪ −1 −1 0 1 1 0 0 0 0 ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ br ⎪ ⎢ −1 −1 0 0 0 0 1 1 0 ⎥ ⎪ 1 ⎥ ⎪ ⎪ ⎢ ⎢ ⎥⎪ ⎪ ⎢ 1 ⎥ (1) (1) cr ⎪ ⎪ ⎪ ⎢ F1 ⎥ ⎪ ⎢ ⎪ ⎥ F 0 F F 0 0 0 0 2 F F ⎢ 1 2 ⎥⎪ 1 2 ⎥ (1) ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥⎪ (2) (2) a ⎪ ⎢ ⎪ ⎥ t F F 0 0 0 0 F F 0 F F ⎢ ⎥ ⎨ (1) ⎬ ⎢ 1 1 2 2 ⎥ 1 2 ⎢ ⎥ ⎥ a 0 2 0 0 0 ⎥ bt F h F h (15) =⎢ ⎢ F1 h2 F2 h2 −2 0 ⎢ 1 2 2 2⎥ b ⎢ ⎪ ⎥⎪ (1) ⎪ ct ⎪ ⎪ ⎢ F1 h1 F2 h1 ⎥ 0 0 0 0 −2 ⎥ ⎪ ⎢ F1 h1 F2 h1 2 0 ⎪ ⎢ ⎥ ⎪ ⎢ ⎥⎪ ⎪ a(2) ⎪ ⎪ ⎢ 0 0 h 0 0 h1 0 0 h2 ⎥ ⎪ 0 ⎥ ⎢ 0 ⎪ ⎢ ⎪ ⎪ ⎥ t ⎢ ⎥⎪ (1) (2) (2) ⎪ (2) ⎪ ⎪ ⎣ k1 F1 k2 F2 ⎦ ⎪ ⎣ −k1 F1 −k2 F2 0 α(1) α2 β α1 α2 −β ⎦ ⎪ ⎪ ⎪ b 1 ⎪ ⎪ t ⎪ ⎪ (1) (1) (2) (2) γ1 γ2 ⎭ γ1 γ2 0 γ1 γ2 0 γ1 γ2 0 ⎩ c(2) t (n) (n) (n) (n) (n) (n) 3 where αm = km Fm (hn /h) , β = 6h1 h2 ke /h3 , γm = hn ikm + Fm (m, n = 1, 2). The reflected and transmitted waves can be symbolically represented in terms of incident wave in the following relation: {a , b , c }T = R3×2 {a, b}T r r r T (1) (1) (1) (1) T a t , b t , ct = T 3×2 {a, b} (16) T (2) (2) (2) (2) T a t , b t , ct = T 3×2 {a, b} w0 = w1 = w2 ,
(1)
(2)
where, R3×2 , T 3×2 , T 3×2 denote the reflection and the transmission matrices whose components are complex in general. The components of the reflection matrix denote the mode conversion between two flexural modes and one extensional mode. For example, R12 means the second incident flexural mode is converted into the first flexural mode reflecting from the tip of the crack. Same as the transmission matrix, the transmitted mode conversion also occurs. There is an extensional mode present in the cracked region. This means the flexural wave will partly transform to the extensional mode when the incident wave transmits into the cracked region. The proportion of the extensional mode conversion compared with the flexural mode conversion depends on the incident wave frequency and position of the crack in the thickness direction. This will be discussed in detail in the section of numerical results.
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3.2. Closed Crack The displacements of the left (un-cracked) region are the same as in the open crack case. Denoting the contact pressure between the two sub-beams as p(x, t) in the cracked region, as shown in Fig. 1(b), with the identical transverse displacement, i.e., w ˆ = w1 = w2 , the following governing equations of flexural wave can be derived: ¨ˆ + p κ2 GA1 (w, ˆ xx +ψ1 ,x ) = ρA1 w 2 ˆ,x + ψ1 ) = ρI1 ψ¨1 χI1 ψ1 ,xx −κ GA1 (w 2 κ GA2 (w, ˆ xx +ψ2 ,x ) = ρA2 w ˆ¨ − p 2 ˆ,x + ψ2 ) = ρI2 ψ¨2 χI2 ψ2 ,xx −κ GA2 (w
(17a) (17b) (17c) (17d)
The contact pressure p can be eliminated by combining Eqs.(17a) and (17c), to obtain ¨ˆ κ2 G (Aw, ˆ xx +A1 ψ1 ,x +A2 ψ2 ,x ) = ρAw
(18)
If the displacements are introduced as: ˆ
ˆ ei(kx−ωt) , w ˆ=W
ˆ
ψ1 = Ψ1 ei(kx−ωt) ,
ˆ
ψ2 = Ψ2 ei(kx−ωt)
(19)
Substituting Eqs.(19) into Eqs.(18), (17b), and (17d), dispersion relation can be determined as follows: ⎛⎡ 2 2 ˆ 2 ⎤ ⎡ ⎤⎞ κ cs Ak −iκ2 c2s A1 kˆ −iκ2 c2s A2 kˆ A 0 0 ⎦ − ω 2 ⎣ 0 q12 0 ⎦⎠ = 0 det ⎝⎣ iκ2 c2s kˆ c2l q12 kˆ2 + κ2 c2s (20) 0 2 2 2ˆ 2 2 ˆ2 2 2 0 0 q 0 cl q2 k + κ cs iκ cs k 2 The results give rise to three modes of flexural waves. Also there are two cut-off frequencies which can be obtained by setting the wave number equal to zero in Eq.(20) and can be analytically determined as follows: κcs κcs ωc1 = , ωc2 = (21) q1 q2 where qj = Ij /Aj (j = 1, 2). For an isotropic beam, the transverse wave velocity and longitudinal wave velocity in the un-cracked beam and sub-beams would be identical. When frequencies approach infinity, the group velocities of three flexural modes can be exactly derived according to Eqs.(20) and (7) cC g1 = κcs ,
cC g2 = cl ,
cC g3 = cl
as
ω→∞
(22)
where the superscript C denotes the cracked beam region. The general solution of the positive-going waves in the cracked region can be written as: ˆ
ˆ
ˆ
w ˆ = at eik1 x + bt eik2 x + dt eik3 x ˆ ˆ ˆ ψn = G(n) at eik1 x + G(n) bt eik2 x + G(n) dt eik3 x 1 2 3
(23)
= −iκ2 c2s kˆj /(qn2 ω 2 − c2l qn2 kˆj2 − κ2 c2s ) (j = 1, 2, 3, and n = 1, 2). where G(n) j The extensional waves in the two sub-beams are the same as in the open crack case. There is a (1) (2) total of eight amplitudes ar , br , cr , at , bt , dt , ct , ct which can be determined from the continuity and equilibrium conditions at x = 0, just the same as in Eqs.(14). Substituting Eqs.(11), (12) and (22) together with Eqs.(3) into Eqs.(14), the equilibrium equations can be written in the matrix form ⎫ ⎡ ⎤⎧ ⎡ ⎤ ar ⎪ −1 −1 0 1 1 1 0 0 ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ br ⎪ ⎢ F1 F2 ⎥ ⎪ ⎢ F1 F2 0 G(1) G(1) G(1) 0 0 ⎥ ⎪ 1 2 3 ⎪ ⎢ ⎪ ⎥⎪ ⎢ ⎥ ⎪ ⎪ (2) (2) (2) ⎪ ⎢ F1 F2 ⎥ ⎪ ⎥ ⎢ F1 c r F2 0 G1 G2 G3 0 0 ⎥ ⎪ ⎪ ⎪ ⎢ ⎪ ⎢ ⎥ ⎨ at ⎬ ⎢ ⎢ F1 h2 F2 h2 −2 0 F1 h2 F2 h2 ⎥ 0 0 2 0 ⎥ ⎢ ⎥ ⎢ ⎥ a (24) =⎢ ⎥ ⎢ F1 h1 F2 h1 2 0 ⎥ b t F h F h 0 0 0 −2 ⎥ ⎪ ⎪ ⎢ 1 1 2 1⎥ b ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎢ 0 0 ⎥ 0 h 0 0 0 h1 h2 ⎥ ⎪ dt ⎪ ⎪ ⎢ ⎢ 0 ⎥⎪ ⎢ ⎥ ⎪ (1) ⎪ ⎪ ⎪ ⎣ ⎣ −k1 F1 −k2 F2 0 υ1 υ2 υ3 β −β ⎦ ⎪ k c F k ⎪ t ⎪ ⎪ 1 1 2 F2 ⎦ ⎪ ⎩ (2) ⎭ γ1 γ2 γ1 γ2 0 ν1 ν2 ν3 0 0 ct
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3 3 (2) ˆj + G(1) + h2 ikˆj + G(2) (j = 1, 2, 3). , ν i k where υj = kˆj G(1) (h /h) + G (h /h) = h 1 2 j 1 j j j j The reflected and transmitted waves are related to the incident wave via the following relation: T (1) (2) T T T {ar , br , cr } = R3×2 {a, b} , at , bt , dt , ct , ct = T 5×2 {a, b} (25) where R3×2 , T 5×2 denote the reflection and the transmission matrices respectively. For the fully contacted crack surfaces, the transmission matrix is only one which contains three coupled flexural modes and two extensional modes in two sub-beams respectively. And the mode conversion of transmitted wave is different from that in the open crack case.
IV. POWER REFLECTION AND TRANSMISSION The wave continuously transports energy as it propagates, the rate of transport into one end of a cross section of the beam being, on average, equal to the rate out of the other end. This speed of transport, or power flow, is given by the rate of work of these forces acting on a beam cross section. Over a period t0 , the average power flow P (per unit length along the wave front) in Timoshenko beam is given by 1 t0 P = − V w˙ + M ψ˙ + N u˙ dt (26) t0 0 where t0 = 2π/ω. There are both incident and reflected waves in the un-cracked region. Since only flexural wave is excited at the far field, the third term of Eq.(26) should vanish. Substituting Eqs.(11) together with the force-displacement relations Eqs.(3) into Eq.(26), the timeaveraged power flow of the incident wave over a cycle can be written as 2
2
Pi = λ1 |a| + λ2 |b| H(ω − ωc ) (27) ! 2 where λm = ωh κ2 Gkm + χq 2 km |Fm | + κ2 GIm (Fm ) 2 (m = 1, 2). The Heaviside function H signifies that the term associated with evanescent wave does not carry energy when ω < ωc . In this case, only the first term of Eq.(27) remains. For the reflected wave in the un-cracked region, the total power flow of reflected wave can be written as follows: 2 2 2 Pr = λ1 |ar | + λ2 |br | H(ω − ωc ) + λe |cr | (28) where λe = ωke χh/2. For a semi-infinite crack, the incident waves transmit waves in the cracked region. In the open cracked region, each sub-beam can be considered as a separate single beam with its own stiffness and thickness. Thus, power flow of transmitted waves in the open cracked region can be expressed as " "2 " "2 " " " (n) "2 (n) " (n) " (n) " (n) " (n) Pt open = λ1 "at " + λ2 "bt " H(ω − ωc ) + λe(n) "ct " (n = 1, 2) (29) # where
(n) λm
= ωhn κ
2
(n) Gkm
+
(n) χqn2 km
" " $% " (n) "2 (n) (n) 2 2, λe = ωke χhn /2 (m, n = 1, 2). "Fm " + κ GIm Fm
The superscripts 1 and 2 denote the two separated sub-beams in the open cracked region. In the closed cracked region, two sub-beams are constrained with identical transverse displacement. Similarly, the total power flow of transmitted wave in the closed cracked region can be written as follows: " " " (n) "2 (n) (n) (n) (n) 2 2 2 Pt closed = η1 |at | + η2 |bt | H(ω − ωc1 ) + η3 |dt | H(ω − ωc2 ) + λe(n) "ct " (30) (n)
where ηj
# " "2 $% " " 2 (n) 2 (j = 1, 2, 3, n = 1, 2). The super= ωhn κ2 Gkˆj + kˆj χqn2 "G(n) + κ GIm G " j j
scripts 1 and 2 denote the two coupled sub-beams in the closed cracked region. The ratios of the reflected and transmitted power (energy) over the incident energy can be defined as (n) (n) Pt open Pt closed Pr (n) (n) R= , Topen , Tclosed = (n = 1, 2) (31) = Pi Pi Pi
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Energy conservation implies R+
2
T (n) = 1
(32)
n=1
For transmitted waves in the cracked region, power (energy) ratios with each mode can be described " "2 " " "2 " "2 "2 (1) " (1) " (2) " (2) " (1) " (1) " (2) " (2) " λ1 "at " + λ1 "at " λ2 "bt " + λ2 "bt " A0 A1 Topen = , Topen = (2) Pt (1) + P Pt (1) + Pt (2) t open open " "2 " "2 open open (1) " (1) " (2) " (2) " (1) (2) 2 |at | λe "ct " + λe "ct " η1 + η1 A0 s0 Topen = , Tclosed = (1) (2) (1) (2) P P t closed + P (33) t open t closed t open + P (1) (2) (1) (2) |bt |2 |dt |2 η2 + η2 η3 + η3 A1 A2 Tclosed = , Tclosed = (1) (2) (1) (2) Pt closed + Pt closed Pt closed + Pt closed " "2 " "2 (1) " (1) " (2) " (2) " λe "ct " + λe "ct " s0 Tclosed = (1) (2) Pt closed + Pt closed where superscripts A0 , A1 , A2 and S0 denote terms related to different propagating wave modes .
as
V. NUMERICAL RESULTS An aluminum beam is chosen as an example in this section. The cross section is rectangular. Young’s modulus E = 72 GPa, and Poisson’s ratio ν = 0.3. The dispersion relations of the un-cracked region of the beam and closed cracked region obtained according to §II and §III are shown in Fig.2, where the crack is located at h1 /h = 0.4. The superscripts 0 and C in the figure denote the uncracked and cracked region of the beam respectively. It can be concluded from Fig.2 that the dispersive curves of two A0 modes are approximately close when the frequency is beyond the cut-off frequency. However, there is a degradation of cracked A0 mode compared with the un-cracked one when the frequencies are below ωc . It is because of the Fig. 2 Group velocities of flexural modes. reduction of the bending stiffness in the cracked region. In addition, both of these two A0 modes tend to reach the transverse wave velocity multiplied by the shear correction factor (κcs ) as the frequency approaches infinity. There is an additional mode in the cracked region which makes the cut-off frequency larger than in the un-cracked region as the split beam is thinner than before. As the frequency becomes higher, the dispersive curves of two spilt modes tend to reach the same value (cl ) as in the un-cracked region. Using Eqs.(31), the ratios of reflected and transmitted power into the incident power are shown in Fig.3(a). In this numerical simulation, all of the results are verified and obey energy conservation. In addition, the two cases have very close results. A slight difference in transmission ratio occurs in the lower frequency domain for these two types of beams. Moreover, there is a high power reflection when the frequency is just beyond the cut-off frequency of the un-cracked region. This is because in this frequency domain the wave propagating into the two sub-beams are still not beyond the cut-off frequencies of thinner sub-beams. There is just one propagating mode transmitted into the cracked region, so all the energy of second flexural mode is reflected back into the un-cracked region. There is almost no extensional wave in the reflected wave, but in the transmitted wave, the conversion of both transmitted extensional wave and flexural wave varying with the frequency can be obtained.
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Fig. 3. (a) Reflected and transmitted energy ratios and (b) Transmitted energy ratio among various wave modes at h1 /h = 0.4.
Figure 3(b) shows the transmitted wave power distributed into different modes. It can be concluded that the A0 mode has most of the energy when ω < ωc . While ω > ωc , the extensional mode takes advantage of the power distribution. It can be considered that the extensional motion in the two sub-beams is mostly converted from ψ(x, t), rotation of the cross section of the beam. The other two flexural modes will be present when the frequency is beyond the two cut-off frequencies of the sub-beams respectively. However, there is no third flexural mode in the open crack case. So when ω > ωc2 , power carried by A1 mode in the open case is equal to the summation of A1 and A2 modes in the closed crack case. In other frequency domains, all the curves resulting from two cases match well. Figure 4(a) shows the effect of various crack positions in the thickness direction on the reflection energy ratio at a fixed frequency ω = 0.1ωc . At this frequency, the transmitted wave only contains the S0 mode and A0 modes, and the transmitted energy ratio for extensional wave S0 power versus crack position is shown in Fig.4(b). It should be noticed that, because of the symmetry of the crack position, the abscissa of curves in Figs.4 is from 0 to 0.5. i.e., h1 /h ∈ (0, 0.5]. The reflection energy ratios of two
Fig. 4. (a) Reflected energy ratio and (b) Transmitted energy ratio for extensional wave S0 power versus different crack positions when ω = 0.1ωc .
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cases are slightly different when h1 /h is below 0.4, while matching well when h1 /h is close to 0.5. It is because the contact pressure will be smaller when the crack is closer to the mid-plane. Especially, in the case of a symmetric crack, i.e. h1 /h = 0.5, there is no contact pressure, and the results of two cases are identical. Generally, the reflected power increases as the crack position comes closer to the middle plane, and reaches a maximum value at h1 /h = 0.5. This can be considered as the reduction of the bending stiffness in the cracked region, the maximum value occurring at h1 /h = 0.5. The energy ratio of the extensional mode increases monotonically with the increase of h1 /h, and also got the maximum value at h1 /h = 0.5.
VI. COMPARISON WITH RESULTS FROM FINITE ELEMENT METHOD In order to verify the accuracy of the analytical method of power reflection and transmission in a semi-infinite crack, a finite element analysis using MSC software is conducted to model the wave propagation in the cracked beam. Figure 5 shows the schematic of the cracked cantilever beam with the geometry, where l = 100 mm, h = 3.2 mm. Point S1 at the middle plane of the un-cracked region acting as sensors is used to measure the reflected wave combined with the incident wave. S2 and S3 at the middle plane of two sub-beams acting as sensors are used to measure the transmitted waves respectively. It should be noticed that sensors measure both the transverse and longitudinal displacements at the middle plane. The element is chosen as a plane strain 4-node rectangular element. Note that the size of every element depends on the frequency of the guided wave. It must be satisfied that there are more than 8 elements in a wavelength to minimize the numerical distortion.
Fig. 5. The schematic of the beam modeled in FEM.
§IV denotes the power flow of flexural wave depending on not only the amplitude of the displacement, but also the frequency of the wave. However it is impossible to generate a single frequency wave in limited time duration, and a narrow incident wave is excited to examine the interaction of flexural waves with the crack by FEM. Thus, a Fourier integral is needed to calculate the wave response. The incident flexural wave is generated at the left of the cantilever beam by adding a moment excitation at the middle node of the left end of the beam, which can be mathematically described by # $ 5 2πf0 t M (t) = M0 H(t) − H t − sin(2πf0 t) 1 − cos (34) f0 5 This wave packet denotes a five-peaked narrowband signal modulated by a Hanning window, which can ensure the energy concentrated around the central frequency f0 . The excitation signal is chosen such that the central frequency is 50 kHz (0.1fc ). Thus there is only the first flexural mode propagating the power. The wave transmitted displacement will be firstly generated at the left end of the beam, described as ai (t). And the frequency spectrum could be obtained using Fourier transform, and can be written as +∞
Ai (ω) =
−∞
ai (t)e−iωt dt
(35)
The magnitudes of wave packets received by three sensors in frequency domain could be obtained by multiplying the reflection and transmission coefficients on incident wave packet. Meanwhile, there is phase-shift caused by the wave propagation in a distance shown in Fig.5. Then the signals received by each sensor can be written as: AS1 (ω) = Ai (ω) e2ik1 l + R11 Ai (ω) e6ik1 l (1) (1) (1) AS2 (ω) = T11 Ai (ω) e4ik1 l+ik1 l , CS2 (ω) = T31 Ai (ω) e4ik1 l+ike l (2) (2) (2) AS3 (ω) = T11 Ai (ω) e4ik1 l+ik1 l , CS3 (ω) = T31 Ai (ω) e4ik1 l+ike l
(36)
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where AS1 (ω), AS2 (ω) and AS3 (ω) denote the flexural signals received by S1 , S2 , and S3 , respectively and CS2 (ω) and CS3 (ω) represent the extensional signals received respectively by S2 and S3 . For example, AS1 (ω) consists of the incident wave and reflected wave. The incident wave propagating to S1 experiences a phase shift by a distance 2l and the reflected wave by 6l. Since both waves propagate in the un-cracked region, a single wave number k1 is used. Then the signal in time domain can be obtained by using the inverse Fourier transform shown as follows: a(t) =
1 2π
+∞ −∞
A(ω)eiωt dω
(37)
where A(ω) represents AS1 (ω), AS2 (ω), AS3 (ω), CS2 (ω) and CS3 (ω) for brevity, and a(t) is the signal in time domain. The numerical results of the symmetric crack case, leading to the maximum reduction of bending stiffness, are presented for example. The signals received from the FEM nodes compared with analytical results are shown in Fig.6. The analytical results chosen from the open crack case have good agreement with the FEM model. As a result of symmetry, signal received by S3 is the same as S2 , so it only shows the signals received by S1 and S2 for brevity.
Fig. 6. The signals received from the FEM nodes compared with analytical results.
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In order to make a convenient comparison, the signals are non-dimensionalized according to the incident wave. Figure 6(a) denotes the incident wave packet passing by associated with reflected wave packet. A good agreement can be seen between FEM model and theory and the amplitudes of the reflected wave are both about 0.1. The transmitted wave contains both flexural waves and extensional waves in S2 mode, so Figs.6(b) and 6(c) shows both of the signals. The signals obtained by two methods match well. Moreover, the extensional wave arrives at position S2 earlier than the flexural wave. This is because there is a length of distance l to be propagated by the transmitted wave, and the extensional wave speed is higher than flexural wave according to the material property in this paper. There are additional wave packets in transmitted extensional waves of FEM results in Fig.6(c), which are the extensional wave reflected from the right boundary of the cantilever beam. For the dispersive phenomenon, the whole power flow of the analytical method is still needed to consider all the components in the whole frequency domain. Thus, the power flow can be written as: P =
1 4π 2
+∞ −∞
2
λ(ω) |A(ω)| dω
(38)
where λ(ω) can be considered as the coefficient of the power flow dependent on frequency, while A(ω) is the transverse displacement amplitude of the transient wave packet by spectral description. As the FEM model is based on the elasticity beam theory, the expression of power flow is different from Timoshenko theory in Eqs.(29), and it can be written as follows: P = −
1 t0
t0 0
h/2 −h/2
(σx u˙ + τxz w) ˙ dzdt
(39)
where the stresses σx , τxz and displacements u , w are functions of coordinate x, z and time t. In order to calculate the power flow through the section y-z, the value of x is chosen as a constant x0 arbitrarily. As the plane strain problem is discussed in this paper, these variables can be obtained at the nodes of the FEM model with different coordinates z. According to Eqs.(38) and (39), the reflected energy ratios of both analytical method and the FEM can be obtained. Reflected energy ratio and transmitted energy ratio for extensional wave S0 power increase with the depth of crack as shown Fig.7. It can be concluded from the figures that the results of FEM have good agreement with the analytical theory in both the reflected energy ratio and transmitted energy ratio for extensional wave S0 power. In this central frequency, the maximum of reflected energy ratio of 0.01 occurs at the symmetric crack, where the transmitted energy ratio for extensional wave S0 power is maximum as 0.13. The curves are
Fig. 7. (a) Reflected energy ratio and (b) Transmitted energy ratio for extensional wave S0 power versus different positions of the crack (h1 /h).
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approximately linearly varying with the position of the crack, so it may be feasible for identifying the position of the crack in the thickness direction.
VII. CONCLUSIONS Wave reflection and transmission in beams containing a semi-infinite crack are presented for two extreme crack surface conditions: open crack and closed crack respectively. The wave reflection and transmission coefficients depend on both frequency and the position of the crack. The mode conversion coefficients among two flexural wave modes and one extensional wave mode are revealed, too. The transmitted energy among various wave modes gives an apparent result of mode conversion varying with frequency. Numerical results show the conservation of power transport in the whole frequency domain, i.e., the power of the incident wave is equal to the summation of the power of the reflection wave and transmission wave, which shows the validity of the analytical method. The analytical results also have good agreement with the numerical results using FEM, which provides an efficient method of locating the crack in thickness direction.
References [1] Wang,J.T.S., Lou,Y.Y. and Gibby,J.A., Vibration of split beams. Journal of Sound and Vibration, 1982, 84: 491-502. [2] Rosalie,S.C., Vaughan,M., Bremner,A. and Chiu,W.K., Variation in group velocity of Lamb waves as a tool for the detection of delamination in GLARE aluminum plate-like structures. Composite Structures, 2004, 66: 77-86. [3] Yan,Y.J. and Yam,L.H., Detection of delamination damage in composite plates using energy spectrum of structural dynamic responses decomposed by wavelet analysis. Computers and Structures, 2004, 82: 347-358. [4] Lase,Y., Ichchou,M.N. and Jezequel,L., Energy flow analysis of bars and beams: theoretical formulations. Journal of Sound and Vibration, 1996, 192: 281-305. [5] Pan,J., Ming,R., Hansen,C.H. and Cark,R.L., Experimental determination of total vibratory power transmission in an elastic beam. Journal of Acoustic Society of America, 1998, 104: 898-906. [6] Bazer,J. and Burridge,R., Energy partition in the reflection and refraction of plane waves. SIAM Journal of Applied Mathematics, 1978, 34: 78-92. [7] Wang,C.H. and Rose,L.R.F., Wave reflection and transmission in beams containing delamination and inhomogeneity. Journal of Sound and Vibration, 2003, 264: 851-872.
ISSN 0894-9166
POWER REFLECTION AND TRANSMISSION IN BEAM STRUCTURES CONTAINING A SEMI-INFINITE CRACK Li Zhou
Wanchun Yuan
(College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China)
Received 8 December 2006; revision received 8 January 2008
ABSTRACT Wave reflection and transmission in a beam containing a semi-infinite crack are studied analytically based on Timoshenko beam theory. Two kinds of crack surface conditions: non-contact (open) and fully contact (closed) cracks, are considered respectively for an isotropic beam. The analytical solution of reflection and transmission coefficients for a semi-infinite crack is obtained. The power reflection and transmission ratios depend on both the frequency and the position of the crack. Numerical results show the conservation of power transport. The transmitted energy among various wave modes is also investigated. A finite element method is used to verify the validity of the analytical results.
KEY WORDS reflection and transmission, crack, power transport, structural health monitoring
I. INTRODUCTION One of the commonly encountered types of defect or damage in structures is crack. The crack not only causes reduction in stiffness but also affects the strength and integrity of the structure, leading to its final failure. The crack also affects vibration and stability characteristics of beam type structures. Vibrations of cracked beams has been investigated[1] . The vibration characteristics of the split beam, such as natural frequencies and mode shapes, have been examined. As expected, the change of these modal parameters is insensitive to the extent of the damage for the first few natural frequencies. When the transient waves propagate in the beam containing delamination[2, 3] , the damaged region generates reflection and transmission waves which carry information on the nature of the damage. Obviously the transmitted flexural wave velocity in the delaminated region is decreased due to the reduction of overall bending stiffness. None of the studies considered the power transport of transient wave packet, which can be useful for damage identification. Power flow in beam-like structures can be obtained from both theory and experiments[4, 5] . To our best knowledge, no power relationship between the incident wave and the reflected or transmitted waves has been presented. Bazer and Burridge[6] derived a general solution of power flow of the plane wave at an interface in a three-dimensional medium. They were concerned with the energy balance associated with the reflection and refraction of monochromatic plane waves governed by the differential equations at a plane interface or boundary. The power flow in the incident wave is equal to the sum of those in the reflected wave and refraction wave, indicating power conservation. Wang and Rose[7] investigated the wave propagation in beams containing delamination and inhomogeneity. The closed delamination form was only considered, and the results of symmetric Project supported by the National Natural Science Foundation of China (Nos. 50478037 and 10572058) and the Research Foundation for the Doctoral Program of Higher Education (No. 20050287016).
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delamination were given in the paper to compare with the results of inhomogeneity. In this study, power reflected from and transmitted through a crack is proposed to analyze the wave behavior in the split beam-like structure. The purpose of this research is to present an analytical method to determine the power reflected from and transmitted through a semi-infinite crack beam. The beam is modeled using Timoshenko beam theory. To model the cracked region, two extreme cases of crack surface conditions during the wave propagation are considered. One is that the crack surfaces are completely open, which means that there is no contact between two sub-beams. The transmitted waves in sub-beams are independent of each other. The other is that the crack surfaces are completely closed, implying there is a contact pressure between the surfaces. These two cases are shown in Figs.1(a) and 1(b), respectively. The two cases are described as open and closed cracks respectively.
Fig. 1. Transient wave propagation in beams containing a semi-infinite crack.
The sections are divided as follows. Dispersion relations of the open and closed cracks are analyzed in §II. The reflection and transmission coefficients are derived in §III. In §IV the power reflection and transmission are analyzed. Numerical results are provided in §V. The comparison of results from analytical solution and finite element method is made in §VI. Finally conclusions are drawn in §VII.
II. DISPERSION RELATION In the beam model the centroidal axis is taken on the x-axis. Based on Timoshenko beam theory, the general displacements of the beam can be described as U (x, z, t) = u(x, t) + zψ(x, t),
W (x, z, t) = w(x, t)
(1)
where u(x, t) is the axial displacement of the beam in the x-direction; ψ(x, t) is the rotation of the cross section of the beam about the y-axis; w(x, t) is the transverse displacement of the beam in the z-direction. The equilibrium equations for the isotropic beam can be written as N,x = ρA¨ u,
V,x = ρAw, ¨
M,x − V = ρI ψ¨
(2)
where N = χAu,x , V = κ2 GA(w,x + ψ), M = χIψ,x (3) and N , V , and M are the axial force, shear force, and bending moment per unit beam width of the beam; χ = E(1 − μ)/[(1 + μ)(1 − 2μ)], E and G are Young’s modulus and shear modulus respectively; κ2 = π 2 /12 is the transverse shear correction factor; A and I are the cross-sectional area and mass moment of inertia of the beam, and ρ is the mass density. For plane wave solutions the displacements can be represented by u = U0 ei(kx−ωt) ,
w = W0 ei(kx−ωt) ,
ψ = Ψ0 ei(kx−ωt)
There are four roots of flexural wave numbers ⎡ ⎤ 2 2 1/2 2 2 c 1 ω c c 1 l 1 + 2l 2 ± 1 − 2l 2 ⎦ + kj = ⎣ 2 κ cs qω 4 κ cs cl
(4)
(j = 1, 2)
(5)
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and k3 − k1 , k4 = −k2 . In the above, cl = χ/ρ, cs = G/ρ, q = I/A. In a relatively low frequency range, the Timoshenko beam experiences a pair of propagating waves (one positive-going and one negative-going) plus two evanescent (near-field) waves, with k purely imaginary. The pair of waves is called the lowest fundamental flexural A0 wave modes. These two near-field waves can be regarded as positive- and negative-going attenuating waves which decay exponentially. Beyond a cut-off frequency, two pairs of propagating waves (two positive-going and two negative-going) co-exist, called A0 and A1 modes. Thus, with an increase of frequency, a non-propagating mode can decay more slowly and eventually becomes a propagating mode. The cut-off frequency can be simply determined by setting k2 = 0, as κcs (6) ωc = q Phase velocities and group velocities are given by cp =
ω , k
cg =
dω dk
(7)
When frequencies approach infinity, the wave velocity becomes non-dispersive. The group velocities of two flexural modes can be simply derived as cg1 = κcs ,
cg2 = cl
as
ω→∞
(8)
A general solution of Eq.(4) is given by w(x, t) =
4
j=1
aj ei(kj x−ωt) ,
ψ(x, t) =
4
Fj aj ei(kj x−ωt)
(9)
j=1
where Fj = (ω 2 − κ2 c2s kj2 )/(iκ2 c2s kj ) (j = 1, 2, 3, 4) and the amplitudes aj may be complex. For the extensional wave mode which is non-dispersive based on the Timoshenko beam theory, the wave number is given by ω ke = (10) cl
III. REFLECTION AND TRANSMISSION COEFFICIENTS It is well known that a propagating wave is incident upon a discontinuity, which gives rise to wave reflection from and transmission through the discontinuity whose amplitudes and phases are given by reflection and transmission coefficients. The incident wave induces propagating and non-propagating evanescent near-field waves. In this section, a propagating flexural wave at far field excited on a beam incident upon a semi-infinite crack is studied. Two extreme cases of crack surface conditions: non-contact (open) and fully contact (closed) cracks, are considered respectively. The reflection and transmission coefficients of both open and closed cracks are derived separately. 3.1. Open Crack Considering a slender beam containing a semi-infinite crack shown in Fig.1(a), the origin of the coordinate is located at the tip of the crack. The beam can be divided into two regions. The left is the un-cracked region which contains both the incident wave and reflected wave, while the right region is the split beam. Since the crack surfaces are open, there is no contact pressure between the surfaces. Only positive-going wave exists in the two sub-beams, which is referred to as the transmitted wave. In the left region (x ≤ 0), it includes the positive-going incident waves and negative-going reflected waves. It can be written as w0 = aeik1 x + beik2 x + ar e−ik1 x + br e−ik2 x ψ0 = F1 aeik1 x + F2 beik2 x − F1 ar e−ik1 x − F2 br e−ik2 x
(11)
When the crack is not located on the mid-plane, an extensional wave is reflected from and transmitted into two split beam regions. The negative-going reflected extensional wave in the left part can be expressed as u0 = cr e−ike x (12)
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where the time dependence term e−iωt has been suppressed here and hereafter. When the excitation frequency is below the cut-off frequency, i.e., ω < ωc , the second wave mode is evanescent since the wave number k2 is purely imaginary. The sign of k2 is chosen as negative to ensure the finiteness of the amplitude as x → −∞. In the right region (x ≥ 0), the displacement of two sub-beams are denoted by u1 , w1 , ψ1 , u2 , w2 , ψ2 . The subscripts 1 and 2 denote the upper and lower sub-beams respectively. As there are only transmitted waves in this region, the general solution of displacement can be written as (n)
(n)
wn = at eik1 x + bt eik2 x (n) (n) (n) (n) (n) (n) ψn = F1 at eik1 x + F2 bt eik2 x (n) (n) un = ct eike x (n)
(n)
(13)
where index n = 1, 2, denoting the parameters associated with each sub-beam. These parameters can be readily obtained by substituting the stiffness and moment of inertia pertaining to the sub-beams in Eqs.(3). The wave number of the extensional wave is independent of beam thickness according to (1) (2) Eqs.(13), implying ke = ke = ke . (1)
(1)
(1)
(2)
(2)
(2)
are The amplitudes of these waves involving nine unknowns ar , br , cr , at , bt , ct , at , bt , ct given by reflection and transmission coefficients which can be determined from the following continuity and equilibrium conditions at the crack tip, x = 0. h2 h1 ψ0 = ψ1 = ψ2 , u1 = u0 + ψ0 , u2 = u0 − ψ0 2 2 (14) h2 h1 N0 = N1 + N2 , M0 = M1 + M2 + N1 − N2 , V0 = V1 + V2 2 2 Substituting Eqs.(11)-(13) together with Eqs.(3) into Eqs.(14), the equilibrium equations can be written in the matrix form, ⎡ ⎤⎧ a ⎫ ⎡ ⎤ r ⎪ −1 −1 0 1 1 0 0 0 0 ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ br ⎪ ⎢ −1 −1 0 0 0 0 1 1 0 ⎥ ⎪ 1 ⎥ ⎪ ⎪ ⎢ ⎢ ⎥⎪ ⎪ ⎢ 1 ⎥ (1) (1) cr ⎪ ⎪ ⎪ ⎢ F1 ⎥ ⎪ ⎢ ⎪ ⎥ F 0 F F 0 0 0 0 2 F F ⎢ 1 2 ⎥⎪ 1 2 ⎥ (1) ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ ⎥⎪ (2) (2) a ⎪ ⎢ ⎪ ⎥ t F F 0 0 0 0 F F 0 F F ⎢ ⎥ ⎨ (1) ⎬ ⎢ 1 1 2 2 ⎥ 1 2 ⎢ ⎥ ⎥ a 0 2 0 0 0 ⎥ bt F h F h (15) =⎢ ⎢ F1 h2 F2 h2 −2 0 ⎢ 1 2 2 2⎥ b ⎢ ⎪ ⎥⎪ (1) ⎪ ct ⎪ ⎪ ⎢ F1 h1 F2 h1 ⎥ 0 0 0 0 −2 ⎥ ⎪ ⎢ F1 h1 F2 h1 2 0 ⎪ ⎢ ⎥ ⎪ ⎢ ⎥⎪ ⎪ a(2) ⎪ ⎪ ⎢ 0 0 h 0 0 h1 0 0 h2 ⎥ ⎪ 0 ⎥ ⎢ 0 ⎪ ⎢ ⎪ ⎪ ⎥ t ⎢ ⎥⎪ (1) (2) (2) ⎪ (2) ⎪ ⎪ ⎣ k1 F1 k2 F2 ⎦ ⎪ ⎣ −k1 F1 −k2 F2 0 α(1) α2 β α1 α2 −β ⎦ ⎪ ⎪ ⎪ b 1 ⎪ ⎪ t ⎪ ⎪ (1) (1) (2) (2) γ1 γ2 ⎭ γ1 γ2 0 γ1 γ2 0 γ1 γ2 0 ⎩ c(2) t (n) (n) (n) (n) (n) (n) 3 where αm = km Fm (hn /h) , β = 6h1 h2 ke /h3 , γm = hn ikm + Fm (m, n = 1, 2). The reflected and transmitted waves can be symbolically represented in terms of incident wave in the following relation: {a , b , c }T = R3×2 {a, b}T r r r T (1) (1) (1) (1) T a t , b t , ct = T 3×2 {a, b} (16) T (2) (2) (2) (2) T a t , b t , ct = T 3×2 {a, b} w0 = w1 = w2 ,
(1)
(2)
where, R3×2 , T 3×2 , T 3×2 denote the reflection and the transmission matrices whose components are complex in general. The components of the reflection matrix denote the mode conversion between two flexural modes and one extensional mode. For example, R12 means the second incident flexural mode is converted into the first flexural mode reflecting from the tip of the crack. Same as the transmission matrix, the transmitted mode conversion also occurs. There is an extensional mode present in the cracked region. This means the flexural wave will partly transform to the extensional mode when the incident wave transmits into the cracked region. The proportion of the extensional mode conversion compared with the flexural mode conversion depends on the incident wave frequency and position of the crack in the thickness direction. This will be discussed in detail in the section of numerical results.
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3.2. Closed Crack The displacements of the left (un-cracked) region are the same as in the open crack case. Denoting the contact pressure between the two sub-beams as p(x, t) in the cracked region, as shown in Fig. 1(b), with the identical transverse displacement, i.e., w ˆ = w1 = w2 , the following governing equations of flexural wave can be derived: ¨ˆ + p κ2 GA1 (w, ˆ xx +ψ1 ,x ) = ρA1 w 2 ˆ,x + ψ1 ) = ρI1 ψ¨1 χI1 ψ1 ,xx −κ GA1 (w 2 κ GA2 (w, ˆ xx +ψ2 ,x ) = ρA2 w ˆ¨ − p 2 ˆ,x + ψ2 ) = ρI2 ψ¨2 χI2 ψ2 ,xx −κ GA2 (w
(17a) (17b) (17c) (17d)
The contact pressure p can be eliminated by combining Eqs.(17a) and (17c), to obtain ¨ˆ κ2 G (Aw, ˆ xx +A1 ψ1 ,x +A2 ψ2 ,x ) = ρAw
(18)
If the displacements are introduced as: ˆ
ˆ ei(kx−ωt) , w ˆ=W
ˆ
ψ1 = Ψ1 ei(kx−ωt) ,
ˆ
ψ2 = Ψ2 ei(kx−ωt)
(19)
Substituting Eqs.(19) into Eqs.(18), (17b), and (17d), dispersion relation can be determined as follows: ⎛⎡ 2 2 ˆ 2 ⎤ ⎡ ⎤⎞ κ cs Ak −iκ2 c2s A1 kˆ −iκ2 c2s A2 kˆ A 0 0 ⎦ − ω 2 ⎣ 0 q12 0 ⎦⎠ = 0 det ⎝⎣ iκ2 c2s kˆ c2l q12 kˆ2 + κ2 c2s (20) 0 2 2 2ˆ 2 2 ˆ2 2 2 0 0 q 0 cl q2 k + κ cs iκ cs k 2 The results give rise to three modes of flexural waves. Also there are two cut-off frequencies which can be obtained by setting the wave number equal to zero in Eq.(20) and can be analytically determined as follows: κcs κcs ωc1 = , ωc2 = (21) q1 q2 where qj = Ij /Aj (j = 1, 2). For an isotropic beam, the transverse wave velocity and longitudinal wave velocity in the un-cracked beam and sub-beams would be identical. When frequencies approach infinity, the group velocities of three flexural modes can be exactly derived according to Eqs.(20) and (7) cC g1 = κcs ,
cC g2 = cl ,
cC g3 = cl
as
ω→∞
(22)
where the superscript C denotes the cracked beam region. The general solution of the positive-going waves in the cracked region can be written as: ˆ
ˆ
ˆ
w ˆ = at eik1 x + bt eik2 x + dt eik3 x ˆ ˆ ˆ ψn = G(n) at eik1 x + G(n) bt eik2 x + G(n) dt eik3 x 1 2 3
(23)
= −iκ2 c2s kˆj /(qn2 ω 2 − c2l qn2 kˆj2 − κ2 c2s ) (j = 1, 2, 3, and n = 1, 2). where G(n) j The extensional waves in the two sub-beams are the same as in the open crack case. There is a (1) (2) total of eight amplitudes ar , br , cr , at , bt , dt , ct , ct which can be determined from the continuity and equilibrium conditions at x = 0, just the same as in Eqs.(14). Substituting Eqs.(11), (12) and (22) together with Eqs.(3) into Eqs.(14), the equilibrium equations can be written in the matrix form ⎫ ⎡ ⎤⎧ ⎡ ⎤ ar ⎪ −1 −1 0 1 1 1 0 0 ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ br ⎪ ⎢ F1 F2 ⎥ ⎪ ⎢ F1 F2 0 G(1) G(1) G(1) 0 0 ⎥ ⎪ 1 2 3 ⎪ ⎢ ⎪ ⎥⎪ ⎢ ⎥ ⎪ ⎪ (2) (2) (2) ⎪ ⎢ F1 F2 ⎥ ⎪ ⎥ ⎢ F1 c r F2 0 G1 G2 G3 0 0 ⎥ ⎪ ⎪ ⎪ ⎢ ⎪ ⎢ ⎥ ⎨ at ⎬ ⎢ ⎢ F1 h2 F2 h2 −2 0 F1 h2 F2 h2 ⎥ 0 0 2 0 ⎥ ⎢ ⎥ ⎢ ⎥ a (24) =⎢ ⎥ ⎢ F1 h1 F2 h1 2 0 ⎥ b t F h F h 0 0 0 −2 ⎥ ⎪ ⎪ ⎢ 1 1 2 1⎥ b ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎢ 0 0 ⎥ 0 h 0 0 0 h1 h2 ⎥ ⎪ dt ⎪ ⎪ ⎢ ⎢ 0 ⎥⎪ ⎢ ⎥ ⎪ (1) ⎪ ⎪ ⎪ ⎣ ⎣ −k1 F1 −k2 F2 0 υ1 υ2 υ3 β −β ⎦ ⎪ k c F k ⎪ t ⎪ ⎪ 1 1 2 F2 ⎦ ⎪ ⎩ (2) ⎭ γ1 γ2 γ1 γ2 0 ν1 ν2 ν3 0 0 ct
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3 3 (2) ˆj + G(1) + h2 ikˆj + G(2) (j = 1, 2, 3). , ν i k where υj = kˆj G(1) (h /h) + G (h /h) = h 1 2 j 1 j j j j The reflected and transmitted waves are related to the incident wave via the following relation: T (1) (2) T T T {ar , br , cr } = R3×2 {a, b} , at , bt , dt , ct , ct = T 5×2 {a, b} (25) where R3×2 , T 5×2 denote the reflection and the transmission matrices respectively. For the fully contacted crack surfaces, the transmission matrix is only one which contains three coupled flexural modes and two extensional modes in two sub-beams respectively. And the mode conversion of transmitted wave is different from that in the open crack case.
IV. POWER REFLECTION AND TRANSMISSION The wave continuously transports energy as it propagates, the rate of transport into one end of a cross section of the beam being, on average, equal to the rate out of the other end. This speed of transport, or power flow, is given by the rate of work of these forces acting on a beam cross section. Over a period t0 , the average power flow P (per unit length along the wave front) in Timoshenko beam is given by 1 t0 P = − V w˙ + M ψ˙ + N u˙ dt (26) t0 0 where t0 = 2π/ω. There are both incident and reflected waves in the un-cracked region. Since only flexural wave is excited at the far field, the third term of Eq.(26) should vanish. Substituting Eqs.(11) together with the force-displacement relations Eqs.(3) into Eq.(26), the timeaveraged power flow of the incident wave over a cycle can be written as 2
2
Pi = λ1 |a| + λ2 |b| H(ω − ωc ) (27) ! 2 where λm = ωh κ2 Gkm + χq 2 km |Fm | + κ2 GIm (Fm ) 2 (m = 1, 2). The Heaviside function H signifies that the term associated with evanescent wave does not carry energy when ω < ωc . In this case, only the first term of Eq.(27) remains. For the reflected wave in the un-cracked region, the total power flow of reflected wave can be written as follows: 2 2 2 Pr = λ1 |ar | + λ2 |br | H(ω − ωc ) + λe |cr | (28) where λe = ωke χh/2. For a semi-infinite crack, the incident waves transmit waves in the cracked region. In the open cracked region, each sub-beam can be considered as a separate single beam with its own stiffness and thickness. Thus, power flow of transmitted waves in the open cracked region can be expressed as " "2 " "2 " " " (n) "2 (n) " (n) " (n) " (n) " (n) Pt open = λ1 "at " + λ2 "bt " H(ω − ωc ) + λe(n) "ct " (n = 1, 2) (29) # where
(n) λm
= ωhn κ
2
(n) Gkm
+
(n) χqn2 km
" " $% " (n) "2 (n) (n) 2 2, λe = ωke χhn /2 (m, n = 1, 2). "Fm " + κ GIm Fm
The superscripts 1 and 2 denote the two separated sub-beams in the open cracked region. In the closed cracked region, two sub-beams are constrained with identical transverse displacement. Similarly, the total power flow of transmitted wave in the closed cracked region can be written as follows: " " " (n) "2 (n) (n) (n) (n) 2 2 2 Pt closed = η1 |at | + η2 |bt | H(ω − ωc1 ) + η3 |dt | H(ω − ωc2 ) + λe(n) "ct " (30) (n)
where ηj
# " "2 $% " " 2 (n) 2 (j = 1, 2, 3, n = 1, 2). The super= ωhn κ2 Gkˆj + kˆj χqn2 "G(n) + κ GIm G " j j
scripts 1 and 2 denote the two coupled sub-beams in the closed cracked region. The ratios of the reflected and transmitted power (energy) over the incident energy can be defined as (n) (n) Pt open Pt closed Pr (n) (n) R= , Topen , Tclosed = (n = 1, 2) (31) = Pi Pi Pi
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Energy conservation implies R+
2
T (n) = 1
(32)
n=1
For transmitted waves in the cracked region, power (energy) ratios with each mode can be described " "2 " " "2 " "2 "2 (1) " (1) " (2) " (2) " (1) " (1) " (2) " (2) " λ1 "at " + λ1 "at " λ2 "bt " + λ2 "bt " A0 A1 Topen = , Topen = (2) Pt (1) + P Pt (1) + Pt (2) t open open " "2 " "2 open open (1) " (1) " (2) " (2) " (1) (2) 2 |at | λe "ct " + λe "ct " η1 + η1 A0 s0 Topen = , Tclosed = (1) (2) (1) (2) P P t closed + P (33) t open t closed t open + P (1) (2) (1) (2) |bt |2 |dt |2 η2 + η2 η3 + η3 A1 A2 Tclosed = , Tclosed = (1) (2) (1) (2) Pt closed + Pt closed Pt closed + Pt closed " "2 " "2 (1) " (1) " (2) " (2) " λe "ct " + λe "ct " s0 Tclosed = (1) (2) Pt closed + Pt closed where superscripts A0 , A1 , A2 and S0 denote terms related to different propagating wave modes .
as
V. NUMERICAL RESULTS An aluminum beam is chosen as an example in this section. The cross section is rectangular. Young’s modulus E = 72 GPa, and Poisson’s ratio ν = 0.3. The dispersion relations of the un-cracked region of the beam and closed cracked region obtained according to §II and §III are shown in Fig.2, where the crack is located at h1 /h = 0.4. The superscripts 0 and C in the figure denote the uncracked and cracked region of the beam respectively. It can be concluded from Fig.2 that the dispersive curves of two A0 modes are approximately close when the frequency is beyond the cut-off frequency. However, there is a degradation of cracked A0 mode compared with the un-cracked one when the frequencies are below ωc . It is because of the Fig. 2 Group velocities of flexural modes. reduction of the bending stiffness in the cracked region. In addition, both of these two A0 modes tend to reach the transverse wave velocity multiplied by the shear correction factor (κcs ) as the frequency approaches infinity. There is an additional mode in the cracked region which makes the cut-off frequency larger than in the un-cracked region as the split beam is thinner than before. As the frequency becomes higher, the dispersive curves of two spilt modes tend to reach the same value (cl ) as in the un-cracked region. Using Eqs.(31), the ratios of reflected and transmitted power into the incident power are shown in Fig.3(a). In this numerical simulation, all of the results are verified and obey energy conservation. In addition, the two cases have very close results. A slight difference in transmission ratio occurs in the lower frequency domain for these two types of beams. Moreover, there is a high power reflection when the frequency is just beyond the cut-off frequency of the un-cracked region. This is because in this frequency domain the wave propagating into the two sub-beams are still not beyond the cut-off frequencies of thinner sub-beams. There is just one propagating mode transmitted into the cracked region, so all the energy of second flexural mode is reflected back into the un-cracked region. There is almost no extensional wave in the reflected wave, but in the transmitted wave, the conversion of both transmitted extensional wave and flexural wave varying with the frequency can be obtained.
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Fig. 3. (a) Reflected and transmitted energy ratios and (b) Transmitted energy ratio among various wave modes at h1 /h = 0.4.
Figure 3(b) shows the transmitted wave power distributed into different modes. It can be concluded that the A0 mode has most of the energy when ω < ωc . While ω > ωc , the extensional mode takes advantage of the power distribution. It can be considered that the extensional motion in the two sub-beams is mostly converted from ψ(x, t), rotation of the cross section of the beam. The other two flexural modes will be present when the frequency is beyond the two cut-off frequencies of the sub-beams respectively. However, there is no third flexural mode in the open crack case. So when ω > ωc2 , power carried by A1 mode in the open case is equal to the summation of A1 and A2 modes in the closed crack case. In other frequency domains, all the curves resulting from two cases match well. Figure 4(a) shows the effect of various crack positions in the thickness direction on the reflection energy ratio at a fixed frequency ω = 0.1ωc . At this frequency, the transmitted wave only contains the S0 mode and A0 modes, and the transmitted energy ratio for extensional wave S0 power versus crack position is shown in Fig.4(b). It should be noticed that, because of the symmetry of the crack position, the abscissa of curves in Figs.4 is from 0 to 0.5. i.e., h1 /h ∈ (0, 0.5]. The reflection energy ratios of two
Fig. 4. (a) Reflected energy ratio and (b) Transmitted energy ratio for extensional wave S0 power versus different crack positions when ω = 0.1ωc .
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cases are slightly different when h1 /h is below 0.4, while matching well when h1 /h is close to 0.5. It is because the contact pressure will be smaller when the crack is closer to the mid-plane. Especially, in the case of a symmetric crack, i.e. h1 /h = 0.5, there is no contact pressure, and the results of two cases are identical. Generally, the reflected power increases as the crack position comes closer to the middle plane, and reaches a maximum value at h1 /h = 0.5. This can be considered as the reduction of the bending stiffness in the cracked region, the maximum value occurring at h1 /h = 0.5. The energy ratio of the extensional mode increases monotonically with the increase of h1 /h, and also got the maximum value at h1 /h = 0.5.
VI. COMPARISON WITH RESULTS FROM FINITE ELEMENT METHOD In order to verify the accuracy of the analytical method of power reflection and transmission in a semi-infinite crack, a finite element analysis using MSC software is conducted to model the wave propagation in the cracked beam. Figure 5 shows the schematic of the cracked cantilever beam with the geometry, where l = 100 mm, h = 3.2 mm. Point S1 at the middle plane of the un-cracked region acting as sensors is used to measure the reflected wave combined with the incident wave. S2 and S3 at the middle plane of two sub-beams acting as sensors are used to measure the transmitted waves respectively. It should be noticed that sensors measure both the transverse and longitudinal displacements at the middle plane. The element is chosen as a plane strain 4-node rectangular element. Note that the size of every element depends on the frequency of the guided wave. It must be satisfied that there are more than 8 elements in a wavelength to minimize the numerical distortion.
Fig. 5. The schematic of the beam modeled in FEM.
§IV denotes the power flow of flexural wave depending on not only the amplitude of the displacement, but also the frequency of the wave. However it is impossible to generate a single frequency wave in limited time duration, and a narrow incident wave is excited to examine the interaction of flexural waves with the crack by FEM. Thus, a Fourier integral is needed to calculate the wave response. The incident flexural wave is generated at the left of the cantilever beam by adding a moment excitation at the middle node of the left end of the beam, which can be mathematically described by # $ 5 2πf0 t M (t) = M0 H(t) − H t − sin(2πf0 t) 1 − cos (34) f0 5 This wave packet denotes a five-peaked narrowband signal modulated by a Hanning window, which can ensure the energy concentrated around the central frequency f0 . The excitation signal is chosen such that the central frequency is 50 kHz (0.1fc ). Thus there is only the first flexural mode propagating the power. The wave transmitted displacement will be firstly generated at the left end of the beam, described as ai (t). And the frequency spectrum could be obtained using Fourier transform, and can be written as +∞
Ai (ω) =
−∞
ai (t)e−iωt dt
(35)
The magnitudes of wave packets received by three sensors in frequency domain could be obtained by multiplying the reflection and transmission coefficients on incident wave packet. Meanwhile, there is phase-shift caused by the wave propagation in a distance shown in Fig.5. Then the signals received by each sensor can be written as: AS1 (ω) = Ai (ω) e2ik1 l + R11 Ai (ω) e6ik1 l (1) (1) (1) AS2 (ω) = T11 Ai (ω) e4ik1 l+ik1 l , CS2 (ω) = T31 Ai (ω) e4ik1 l+ike l (2) (2) (2) AS3 (ω) = T11 Ai (ω) e4ik1 l+ik1 l , CS3 (ω) = T31 Ai (ω) e4ik1 l+ike l
(36)
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where AS1 (ω), AS2 (ω) and AS3 (ω) denote the flexural signals received by S1 , S2 , and S3 , respectively and CS2 (ω) and CS3 (ω) represent the extensional signals received respectively by S2 and S3 . For example, AS1 (ω) consists of the incident wave and reflected wave. The incident wave propagating to S1 experiences a phase shift by a distance 2l and the reflected wave by 6l. Since both waves propagate in the un-cracked region, a single wave number k1 is used. Then the signal in time domain can be obtained by using the inverse Fourier transform shown as follows: a(t) =
1 2π
+∞ −∞
A(ω)eiωt dω
(37)
where A(ω) represents AS1 (ω), AS2 (ω), AS3 (ω), CS2 (ω) and CS3 (ω) for brevity, and a(t) is the signal in time domain. The numerical results of the symmetric crack case, leading to the maximum reduction of bending stiffness, are presented for example. The signals received from the FEM nodes compared with analytical results are shown in Fig.6. The analytical results chosen from the open crack case have good agreement with the FEM model. As a result of symmetry, signal received by S3 is the same as S2 , so it only shows the signals received by S1 and S2 for brevity.
Fig. 6. The signals received from the FEM nodes compared with analytical results.
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In order to make a convenient comparison, the signals are non-dimensionalized according to the incident wave. Figure 6(a) denotes the incident wave packet passing by associated with reflected wave packet. A good agreement can be seen between FEM model and theory and the amplitudes of the reflected wave are both about 0.1. The transmitted wave contains both flexural waves and extensional waves in S2 mode, so Figs.6(b) and 6(c) shows both of the signals. The signals obtained by two methods match well. Moreover, the extensional wave arrives at position S2 earlier than the flexural wave. This is because there is a length of distance l to be propagated by the transmitted wave, and the extensional wave speed is higher than flexural wave according to the material property in this paper. There are additional wave packets in transmitted extensional waves of FEM results in Fig.6(c), which are the extensional wave reflected from the right boundary of the cantilever beam. For the dispersive phenomenon, the whole power flow of the analytical method is still needed to consider all the components in the whole frequency domain. Thus, the power flow can be written as: P =
1 4π 2
+∞ −∞
2
λ(ω) |A(ω)| dω
(38)
where λ(ω) can be considered as the coefficient of the power flow dependent on frequency, while A(ω) is the transverse displacement amplitude of the transient wave packet by spectral description. As the FEM model is based on the elasticity beam theory, the expression of power flow is different from Timoshenko theory in Eqs.(29), and it can be written as follows: P = −
1 t0
t0 0
h/2 −h/2
(σx u˙ + τxz w) ˙ dzdt
(39)
where the stresses σx , τxz and displacements u , w are functions of coordinate x, z and time t. In order to calculate the power flow through the section y-z, the value of x is chosen as a constant x0 arbitrarily. As the plane strain problem is discussed in this paper, these variables can be obtained at the nodes of the FEM model with different coordinates z. According to Eqs.(38) and (39), the reflected energy ratios of both analytical method and the FEM can be obtained. Reflected energy ratio and transmitted energy ratio for extensional wave S0 power increase with the depth of crack as shown Fig.7. It can be concluded from the figures that the results of FEM have good agreement with the analytical theory in both the reflected energy ratio and transmitted energy ratio for extensional wave S0 power. In this central frequency, the maximum of reflected energy ratio of 0.01 occurs at the symmetric crack, where the transmitted energy ratio for extensional wave S0 power is maximum as 0.13. The curves are
Fig. 7. (a) Reflected energy ratio and (b) Transmitted energy ratio for extensional wave S0 power versus different positions of the crack (h1 /h).
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approximately linearly varying with the position of the crack, so it may be feasible for identifying the position of the crack in the thickness direction.
VII. CONCLUSIONS Wave reflection and transmission in beams containing a semi-infinite crack are presented for two extreme crack surface conditions: open crack and closed crack respectively. The wave reflection and transmission coefficients depend on both frequency and the position of the crack. The mode conversion coefficients among two flexural wave modes and one extensional wave mode are revealed, too. The transmitted energy among various wave modes gives an apparent result of mode conversion varying with frequency. Numerical results show the conservation of power transport in the whole frequency domain, i.e., the power of the incident wave is equal to the summation of the power of the reflection wave and transmission wave, which shows the validity of the analytical method. The analytical results also have good agreement with the numerical results using FEM, which provides an efficient method of locating the crack in thickness direction.
References [1] Wang,J.T.S., Lou,Y.Y. and Gibby,J.A., Vibration of split beams. Journal of Sound and Vibration, 1982, 84: 491-502. [2] Rosalie,S.C., Vaughan,M., Bremner,A. and Chiu,W.K., Variation in group velocity of Lamb waves as a tool for the detection of delamination in GLARE aluminum plate-like structures. Composite Structures, 2004, 66: 77-86. [3] Yan,Y.J. and Yam,L.H., Detection of delamination damage in composite plates using energy spectrum of structural dynamic responses decomposed by wavelet analysis. Computers and Structures, 2004, 82: 347-358. [4] Lase,Y., Ichchou,M.N. and Jezequel,L., Energy flow analysis of bars and beams: theoretical formulations. Journal of Sound and Vibration, 1996, 192: 281-305. [5] Pan,J., Ming,R., Hansen,C.H. and Cark,R.L., Experimental determination of total vibratory power transmission in an elastic beam. Journal of Acoustic Society of America, 1998, 104: 898-906. [6] Bazer,J. and Burridge,R., Energy partition in the reflection and refraction of plane waves. SIAM Journal of Applied Mathematics, 1978, 34: 78-92. [7] Wang,C.H. and Rose,L.R.F., Wave reflection and transmission in beams containing delamination and inhomogeneity. Journal of Sound and Vibration, 2003, 264: 851-872.