藉由懸臂梁動態彎曲探討薄膜材料機械性質之識別---理論及實驗

行政院國家科學委員會補助專題研究計畫

藉由懸臂梁動態彎曲探討薄膜材料機械性質之識別:

理論及實驗

計畫類別：個別型計畫 □整合型計畫 計畫編號：NSC 96-2221-E-006-068

執行期間：96 年 8 月 1 日 至 97 年 7 月 31 日 計畫主持人：王雲哲

計畫參與人員：馮凱文、陳修宗、周鴻儒

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中 華 民 國 九 十 六 年 十 月 十 三 日

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告

1

1

行政院國家科學委員會專題研究計畫期中報告

藉由懸臂梁動態彎曲 探討薄膜材料 機 械性質之識別: 理論及實驗 Identification of mechanical properties of thin-film materials via dynamic bending

of cantilevers: theoretical and experimental development

計畫編號： NSC 96-2221-E-006-068

執行期限：96 年 8 月 1 日 至 97 年 7 月 31 日

主持人： 王雲哲 國立成功大學土木工程系 [email protected]

計畫參與人員： 馮凱文 國立成功大學土木工程系 [email protected]

陳修宗 國立成功大學土木工程系 [email protected] 周鴻儒 國立成功大學土木工程系 [email protected] 一、中英文摘要

金屬奈米多層薄膜已被廣泛的研究，是因為其具有高降伏強度及大塑性變形的能力。本計 畫至今以執行兩個多月，主要以推導共振頻率試驗的理論基礎， 幾何非線性的懸臂樑振動 問題的數學架構已建立。共振頻率儀器為一種新的實驗方法，可用以量測自我支撐薄膜材 料的楊氏模數、線性黏彈性阻尼係數及機械疲勞性質。目前發現幾何非線性效應使得懸臂 梁的振動問題形成達分系統，此系統使得懸臂梁振動頻率成為振幅的函數。有限元素計算 顯示，古典理論與有限元素預測趨勢一致，但因古典理論缺少剪力及軸向的變形，使得古 典理論與有限元素的預測不完全一致。共振超音波頻譜儀已架設完成，測試階段顯示，此 儀器能有效預估材料及剪力模數，測試錫立方塊的結果顯示，錫的剪力模數為19 GPa。

關鍵詞: 薄膜, 幾何非線性, 懸臂樑, 達分振動子, 共振.

Abstract

To identify damping properties of materials, such as loss tangent, which are independent of geometry and experimental methods, we have derived accurate equations of motion to model the behavior of a cantilever beam under large displacements. And, multiple-scale analysis has been adopted to study the weakly nonlinear system. Our model is a generalization of classical elastica theory. Moreover, this model may provide accurate data-reduction method to obtain and scrutinize maximum stress of the cantilever under the loading and clamping conditions with the consideration of the Saint Venant’s principle. The current status of the resonant frequency

2

2

device to measure the mechanical properties of nano-structured, self-supported thin films through based-excitation vibrations is under construction for an experimental chamber and stage.

Key components of electronics for experiment have been obtained. Hence, the resonant ultrasound spectroscopy device has been completed and running. However, Labview programming is still under further development so that signals can be filtered with a software-level lock-in amplifier to reduce noise. The resonant ultrasound spectroscopy showed the Young’s modulus of Sn was about 19 GPa.

Keywords: thin film, geometrical nonlinearity, cantilever beam, Duffing oscillator, resonance

二、報告內容 如附件.

三、參考文獻 如附件.

四、計畫成果自評 如附件.

NSC 96-2221-E-006-068 Midterm Brief Report

Identification of mechanical properties of thin-film materials via dynamic bending of cantilevers: theoretical and experimental

development

Yun-Che Wang, Kevin Foun, H. T. Chen, H. R. Chou

Department of Civil Engineering National Cheng Kung University

Taiwan

1 Abstract

To identify damping properties of materials, such as loss tangent, which are independent of geometry and experimental methods, we have derived accurate equations of motion to model the behavior of a cantilever beam under large displacements. And, multiple-scale analysis has been adopted to study the weakly nonlinear system. Our model is a generalization of classical elastica theory. Moreover, this model may provide accurate data-reduction method to obtain and scrutinize maximum stress of the cantilever under the loading and clamping conditions with the consideration of the Saint Venants principle. The current status of the resonant frequency device to measure the mechanical properties of nano-structured, self-supported thin films through based-excitation vibrations is under construction for an experimental chamber and stage. Key components of electronics for experiment have been obtained. Hence, the resonant ultrasound spectroscopy device has been completed and running. However, Labview programming is still under further development so that signals can be filtered with a software-level lock-in amplifier to reduce noise. Resonant ultrasound spectroscopy showed the Young’s modulus of Sn was about 19 GPa.

Keywords: thin film, geometrical nonlinearity, cantilever beam, Duffing oscillator, resonance

2 Introduction

Mechanical properties of nano-structured thin film are summarized in [9] for its high strength and high ductility. However, most current experimental results of thin film are obtained from nanoindentation. There is a need to cross-check the mechanical properties of thin film with different experimental techniques. A dynamical bending technique is a plausible one.

Conventional beam theories are the (a) exact elasticity equations, (b) Euler-Bernoulli beam theory, and (c) Timoshenko beam theory. All these theories assume linearity. In other words, the change of reference geometry during the deformation process is ignored. A linear beam model would suffice when dealing with small deformations. When the deformations are moderately large, for accurate modeling, several nonlinearities also need to be included. It is impossible to come up with a very general three-dimensional beam theory incorporating all possible nonlinearities and secondary effects, like rotatory inertia, shear deformation, warping, damping, static deformation, etc. However, insignificant nonlinearities and secondary effects are dropped to (a) simplify various expressions, (b) make the model manageable, and (c) facilitate solving the model equations.

Kane et al. [8] developed a comprehensive theory dealing with small vibrations of a beam attached to a base that is performing an arbitrary but prescribed three-dimensional motion (translation and rotation). This theory is applicable to beams with arbitrary cross section and spatially varying material properties. Crespo da Silva et al. [3], [4], [5], [6], [7] studied the static equilibrium deflection and natural frequencies associated with infinitesimally small oscillations about the static equilibrium. Their theoretical results are in good agreement with experimental results. Further derivation on the nonlinear differential equations of motion for a rotating beam are made with the objective of retaining all possible contributions due to cubic nonlinearities. It was identified that the most significant cubic nonlinear terms are those associated with the structural geometric nonlinearity in the torsion equation. Nonlinearities due to curvature, inertia, and extension were accounted for in a mathematically consistent manner. Pai and Nayfeh [12] also developed the nonlinear equations describing the extensional-flexural-flexural-torsional vibrations of slewing or rotating metallic and composite beams. The equations contain structural coupling terms and quadratic and cubic nonlinearities due to curvature and inertia. Further development by the authors used Euler-angle-like rotations are used to relate the deformed and undeformed configurations to include warping and three-dimensional effects. Cusumano and Moon [1], [2] conducted experiments with an externally excited cantilever beam, and they observed a cascading of energy into the low-frequency modes. Recenly, Malatkar [11] reviewed the three-dimensional behavior of a thin cantilever beam, and Xiao et al. [13] studied the dynamics of a viscoelastic Timoshenko beam.

We consider the dynamical system of a 2D cantilever elastica under base excitation in translation and rotation to simulate the cantilever beam driven by a piezoelectric cantilever whose free end is rigidly attached to the base of the elastica. The equations of motion for the elastica are derived, and comparison between theory and experiment is presented. To solve the governing equation, a multi- scale perturbation technique is adopted. Perturbation techniques for solving nonlinear differential are summarized in [10].

3 Material beam theories

Several material beam theories are discussed and derived in tis section. The most suitable formu- lation for the present work is the governing equation of two-dimensional, viscoelastic elastica. In order to solve this equation, a perturbation technique is used, and briefly summarized in the next section. A typical bending schematic of a cantilever beam is shown in Figure 1.

3.1 Euler-Bernoulli beam theory

The strain energy density of the beam, u = u(x, y, z), can be calculated as follows with Hooke’s law, σ = Eε.

u = Z

σdε = Z

Eεdε = 1

2Eε² = 1

2σε. (1)

The total strain energy is the integral of the strain energy density over the whole volume of the beam, and with the curvature-strain relation, ε = −κy,one can obtain

U = Z

Ω

udV = E 2

Z

Ω

ε²dV = E 2

Z

Ω

(−κy)²dV = EI 2

Z

κ²dx. (2)

Here we assume κ = κ(x). It is known that the stress-moment relation is σ = −M y

I . (3)

And, the resultant shear and moment can be calculated from stress as follows.

V = Z

σdA, (4)

M = − Z

yσdA. (5)

The relationship between resultants shears and moments and applied distributed loads is as follows.

∂V

∂x = −q, (6)

∂M

∂x = −V, (7)

∂²M

∂x² = q. (8)

To generalize quasi-static beam theory to viscoelastic one, we use the constitutive relation of the standard linear solid, P (σ) = Q(ε) = −yQ(κ), and then Eqs. (4) and (5) become.

P (V ) = Z

Q(ε)dA, (9)

P (M ) = − Z

yQ(ε)dA = IQ(κ). (10)

Here P and Q are viscoelastic differential operators.

3.2 Timoshenko beam theory

The Timoshenko beam theory is an attempt of generalizing the Euler-Bernoulli, which assumes that the local slope of the beam is composed of two parts: an angle (φ) and a shear deformation (γ).

∂v

∂x = φ + γ. (11)

Since shear force is proportional to shear deformation only, one obtains V = GkA ∂v

∂x − φ



, (12)

where k is the shape factor, largely used in the literature, no theoretical foundation. However, the curvature of the beam is

κ = M

EI = ∂φ

∂x. (13)

From Eqs. (6) and (7), the governing equations of the Timoshenko beam can be expressed as follows.

k(v_xx− φ_x) + q = mv_tt, (14)

EIφ_xx+ k(v_x− φ) = I_ρφ_tt. (15)

Here m = ρA and Iρ = ρI. Combining Eqs. (14) and (15), one reaches a single governing, as follows.

EIv_xxxx− ρI(1 + E

kG)v_xxtt+ ρAv_tt+ρ²I

kGv_tttt = q − EI

kGAq_xx+ ρI

kGAq_tt. (16) To generalize the model to include linear viscoelastic effects, one introduces the following constitu- tive relations.

P₀(σ) = Q₀(ε), (17)

P₁(τ ) = Q₁(γ). (18)

Here P0 and Q0 are differential operators with respect to time for normal stress and strain, and P1

and Q1 for shear ones.

P₀ = ∂

∂t+ τ_ε Q₀= ∂

∂t+ τ_σ (19)

P1 = ∂

∂t+ τγ Q1 = ∂

∂t+ ττ (20)

3.3 Two-dimensional theory of elastica

Force balance of a deformed beam under external distributed applied load F is as follows.

df

ds = −F (21)

Here f is the internal force inside the beam and is generated by the beam to against the external applied force, s is the measuring coordinate along the deformed beam.

Moment balance leads to

dM + dr × f = 0, (22)

dM

ds = f × t, (23)

t = dr

ds (24)

M is the moment generated inside the beam to balance the couples from the internal force f . r is the position vector measuring the beam. Note the beam is not subject to external couples.

Bending moment is related to the torsion angle of the cross section as follows. Assume area moments of inertia, I, are the same along any principal directions.

M = M_ξe_ξ+ Mηeη+ M_ζe_ζ (25)

Ω = Ωξeξ+ Ωηeη+ Ωζeζ (26)

M_ξ = EIΩ_ξ (27)

M_η = EIΩ_η (28)

M_ζ = GJ Ω_ζ (29)

The torsion angle can be related to t and dt/ds as follows, due to Kirchhoff’s kinematic analogue.

dt

ds = Ω × t (30)

(Ω × t) × t = −t × dt

ds (31)

Since (A × B) × C = (A · C)B − A(B · C), Ω = t × dt

ds + (t · Ω)t (32)

Observe that t · Ω = Ω_ζ, twist moment along the longitudinal direction of the beam, M = EIt ×dt

ds + tGJ Ω_ζ (33)

Note that

d

ds(M · t) = GJdΩ_ζ

ds (34)

GJdΩ_ζ

ds = dM

ds · t + M ·dt

ds (35)

GJdΩ_ζ

ds = (f × t) · t + M ·dt

ds (36)

GJdΩ_ζ

ds = M ·dt

ds (37)

GJdΩ_ζ

ds = (EIt ×dt

ds+ tGJ Ωζ) · dt

ds (38)

GJdΩ_ζ

ds = 0 (39)

In other words, Ω_ζ is a constant throughout the beam. If there is no applied twist, the constant is zero. Hence, for the beam under no twisting moment,

M = EIt × dt

ds (40)

M = EIdr ds ×d²r

ds² (41)

Combining with the moment balance equation, EI d

ds

 dr ds ×d²r

ds²



= f ×dr

ds (42)

EI dr ds ×d³r

ds³



= f ×dr

ds (43)

For a cantilever beam under lateral bending by a point load at its free end, f = −f ˆy

r = xˆx + y ˆy (44)

dr

ds = sin θ ˆx + cos θ ˆy (45)

d²r

ds² = cos θdθ

dsx − sin θˆ dθ

dsyˆ (46)

EI d ds

 dr ds ×d²r

ds²



= −EId²θ

ds²z = −f sin θˆˆ z (47) The governing equation on the deformation of the beam in terms of angle with respect to the y-axis is

EId²θ

ds² − f sin θ = 0 (48)

EI 2

 dθ ds

2

+ f cos θ = c₁ (49)

Here c1 = cos θ0 due to the boundary condition at the free end, where s = L, θ = θ0 and M =

−EI^dθ_ds = 0. The relationship between the total length L and θ is as follows. Note θ measures from θ₀ to π/2, and θ₀, the angle of the free end, is an unknown.

L =

s EI 2f

Z π/2 θ0

√ dθ

cos θ0− cos θ (50)

ds = s

EI 2f

√ dθ

cos θ₀− cos θ (51)

The coordinates of each point of the deformed beam, x =R sin θds and y = R cos θds, are

x =

s EI

2f Z θ

θ0

sin θdθ

√cos θ0− cos θ = s

2EI f (p

cos θ0−p

cos θ0− cos θ) (52)

y = s

EI 2f

Z θ θ0

cos θdθ

√cos θ0− cos θ (53)

If the loading is f = −f ˆx, the beam is under compression.

EId²θ

ds² − f cos θ = 0 (54)

EI 2

 dθ ds

2

− f sin θ = c₂ (55)

The boundary condition at the free end leads to c₂= sin θ₀.

L =

s EI

2f Z π/2

θ0

√ dθ

sin θ − sin θ0

(56)

ds = s

EI 2f

√ dθ

sin θ − sin θ₀ (57)

x =

s EI

2f Z _θ

θ0

sin θdθ

√sin θ − sin θ₀ (58)

y =

s EI

2f Z θ

θ0

cos θdθ

√sin θ − cos θ0

(59) Alternatively, by changing the coordinate system,

L =

s EI

2f Z θ0

0

√ dθ

cos θ − cos θ₀ (60)

x =

s EI

2f Z θ

0

sin θdθ

√cos θ − cos θ₀ = s

2EI f (p

1 − cos θ0−p

cos θ − cos θ0) (61)

y =

s EI

2f Z θ

0

cos θdθ

√cos θ0− cos θ (62)

Note when θ0  1, cos θ₀ ≈ 1 −¹₂θ²₀+_4!¹θ₀⁴.

L =

s EI

2f Z θ0

0

dθ

pθ²₀− θ² (63)

L = π

2 s

EI

f (64)

f = π²EI

4L (65)

The Euler buckling solution is therefore re-constructued.

3.4 Viscoelasticity

For a standard linear solid, the constitutive relation between stress and strain is as follows. It is noted that τ_ε and τ_σ are in units of the inverse of time. For the consistency of initial conditions, one requires σ(0) = E0ε(0).

˙σ + τεσ = E0( ˙ε + τσε). (66)

Assuming σ(0) = 0, The solution of Eq. (66) in time domain is σ(t) = E₀ε(t) + E₀(τ_σ− τ_ε)

Z _t

0

εe^{−τε(t−τ )}dτ. (67)

After some rearrangement, one can show that the following equation is equivalent to Eq. (67).

σ(t) = E0ε(t) − Z t

0

ε∂Y

∂τdτ. (68)

Y = E0+ E0(τσ

τε

− 1)(1 − e^−τετ). (69)

From the Boltzmann principle, the general stress-strain relationship can be expressed as follows.

σ = Z t

0

E(t − τ )∂ε

∂τdτ. (70)

After integration by parts and applying ε(0) = 0, one obtains the following.

σ(t) = E(0)ε(t) − Z t

0

ε∂E

∂τ dτ. (71)

By comparing Eq. (68) and Eq. (71), one can conclude that the standard linear solid model is a special case of the general stress-strain relationship under the Boltzmann principle, where the relaxation modulus is stipulated by Eq. (69).

4 Elastica in three dimensions

For a general three-dimentional cantilever beam, the 3-2-1 Euler’s angles (ψ, θ, φ) are adopted to describe its position in space. The Euler’s angles transform coordinates from (x,y,z) to (x⁰,y⁰,z⁰), (x⁰⁰,y⁰⁰,z⁰⁰), and finally to (ξ,η,ζ).

ω = ˙ψ ˆz + ˙θ ˆy⁰+ ˙φ ˆξ = ω_ξξ + ωˆ _ηη + ωˆ _ζζˆ (72) Here

ω_ξ = φ − ˙˙ ψ sin θ (73)

ωη = ψ cos θ sin φ + ˙˙ θ cos φ (74)

ωζ = ψ cos θ cos φ − ˙˙ θ sin φ (75)

By the three-dimensional Kirchhoff’s kinematic analogue, the curvature vector of the deformed beam can be obtained as follows.

ρ(s, t) = ρ_ξξ + ρˆ ηη + ρˆ _ζζˆ (76)

ρ_ξξˆ = φ⁰− ψ⁰sin θ (77)

ρ_ηηˆ = ψ⁰cos θ sin φ + θ⁰cos φ (78)

ρ_ζ = ψ⁰cos θ cos φ − θ⁰sin φ (79)

Alternatively, curvatures can be obtained by the following.

ρξ = ∂ ˆη

∂s · ˆζ (80)

ρη = −∂ ˆξ

∂s · ˆζ (81)

ρζ = ∂ ˆξ

∂s · ˆη (82)

The in-extensibility conditions ensure

(1 + u⁰²)²+ v⁰²+ w⁰²= 1 (83)

For a point on the corss-section (η, ζ), by the definition of Green’s strain tensor, dr_p^∗· dr_p^∗− dr_p· drp = 2(ds dη dξ)[ij](ds dη dξ)^T, where rp = sˆx + η ˆy + ζ ˆz and rp^∗ = (s + u)ˆx + v ˆy + wˆz + η ˆη + ζ ˆζ.

₁₁= ζρ_η− ηρ_ζ (84)

γ₁₂= 2₁₂= −ζρ_ξ (85)

γ13= 213= ηρ_ξ (86)

All other strains are zero.

The Lagrangian of the elastica under certain assumptions is as follows.

L = T − V = Z L

0

`dx. (87)

` = 1

2m( ˙u²+ ˙v²+ ˙w²) +1

2(J_ξω_ξ²+ Jηω²_η+ J_ζω²_ζ) − 1

2(Dξρ²_ξ+ Dηρ²_η + Dζρ²_ζ) +1

2λ[1 − (1 + u⁰)²− v⁰²− w⁰²]. (88) The variation of work done by non-conservative forces can be calculated as follows.

δWnc = Z L

0

[(Qu− c_uu)δu + (Q˙ v− c_v˙v)δv + (Qw− c_ww)δw + (Q˙ _φ− c_φφ)δφ]ds.˙ (89) δW_nc =

Z L 0

[Q^∗_uδu + Q^∗_vδv + Q^∗_wδw + Q^∗_φδφ]ds. (90)

To obtain the equations of motion, one needs to solve the following variational equation.

δI = Z _t2

t1

(δL + δW_nc)dt = Z _t2

t1

Z _L

0

[δ` + Q^∗_uδu + Q^∗_vδv + Q^∗_wδw + Q^∗_φδφ]dsdt = 0. (91)

5 Viscoelastic elastica in two dimensions

It is known that the dynamical system of the elastica under the approximation up to the third order resembles the Duffing system [11]. The convention of symbol usage is as follows. A super dot denotes differentiation with respect to time, and a prime indicates differentiation with respect to space.

The linear theory of the Euler-Bernoulli cantilever beams with base excitation, as shown in the

mathematical system, Eqs. (92) - (97), can be derived in various ways.

L = T − V. (92)

V = 1

2 Z L

0

EIκ²dx. (93)

δV = EI

Z L 0

y⁰⁰δy⁰⁰dx. (94)

δV

EI = [y⁰⁰δy⁰]^L₀ − [y⁰⁰⁰δy⁰]^L_o + Z L

0

y⁰⁰⁰⁰δydx. (95) m(x)∂²y(x, t)

∂t² + EI∂⁴y(x, t)

∂x⁴ = 0. (96)

m(x)∂²y(x, t)

∂t² + EI∂⁴y(x, t)

∂x⁴ = −m(x)∂²R(t)

∂t² . (97)

The geometrical assumptions of the elastica indicate that one must use the exact curvature- deflection relation in Eq. (98) and inextentionality condition in Eq. (99).

κ = v⁰⁰

(1 + v⁰²)^3/2. (98)

(1 + u⁰²)²+ v⁰² = 1. (99)

The kinetic energy of the system is as follows.

T = 1

2 Z L

0

m ˙u²+ ˙v²
dx. (100)

Here m = m(x) is the line density of the beam in units of kg/m. Since u(x, t) is strongly related to v(x, t), to facilitate later derivations, we rewrite Eq. (99) as follows.

u⁰ = −1

2v⁰², or (101)

u = −1 2

Z x 0

v⁰²dx. (102)

The following shows how to obtain δT . δT =

Z t2

t1

Z L 0

m ( ˙uδ ˙u + ˙vδ ˙v) dxdt. (103)

δT = Z L

0

[ ˙uδu]^t_t²₁+ [ ˙vδv]^t_t²₁dx − m Z t2

t1

Z L 0

¨

uδu + ¨vδvdxdt. (104) To convert δu to δv, one needs to use the following identity.

d dx

Z x L

F (x)dxδu



= F (x)δu +

Z x L

F (x)dx



δu⁰, (105)

where

F (x) = −1 2

∂²

∂t² Z _x

0

v⁰²dx. (106)

With the use of Eq. (101), (104) and (105), δT can be expressed as follows.

δT = Z L

0

[ ˙uδu]^t_t²₁+ [ ˙vδv]^t_t²₁dx + Z t2

t1

[(v⁰ Z x

L

F (x)dx)δv]^L₀dt + Z t2

t1

Z L 0

m(¨v − ∂

∂x{v⁰ Z x

L

F (x)dx})δvdxdt. (107)

The potential energy of the system is as follows.

V = EI

2 Z L

0

y⁰⁰²

(1 + y⁰²)³dx. (108)

V = EI

2 Z L

0

y⁰⁰²(1 + b1y⁰²+ b2y⁰⁴)dx = V1+ V2+ V3. (109) V₂ = EIb₁

2 Z L

0

y⁰⁰²y⁰²dx. (110)

δV2 = EIb1

Z L 0

(y⁰⁰y⁰²δy⁰⁰+ y⁰⁰²y⁰δy⁰)dx. (111) δV₂

EIb₁ = [y⁰⁰y⁰²δy⁰]^L₀ − [(y⁰⁰y⁰²)⁰δy]^L₀ + [y⁰⁰²y⁰δy⁰]^L₀ + Z L

0

[(y⁰⁰y⁰²)⁰⁰− (y⁰⁰²y⁰)⁰]δydx.(112) (y⁰⁰y⁰²)⁰⁰− (y⁰⁰²y⁰)⁰ = y⁰⁰⁰⁰y⁰²+ y⁰⁰³+ 4y⁰y⁰⁰y⁰⁰⁰. (113) The equation of motion of a cantilever beam under horizontal base excitation is as follows.

m¨v + EIv⁰⁰⁰⁰+ EIb1(v⁰⁰⁰⁰v⁰²+ v⁰⁰³+ 4v⁰v⁰⁰v⁰⁰⁰) + m

2

∂

∂x

 v⁰

Z x L

 ∂²

∂t² Z x

0

v⁰²dx

 dx



= mabcosΩt. (114)

Here ab indicates the base acceleration. If the material of the beam behaves as the standard linear solid in the context of viscoelasticity, Eq. (114) needs to be modified as follows.

mP (¨v) + Q Iv⁰⁰⁰⁰+ α v⁰⁰⁰⁰v⁰²+ v⁰⁰³+ 4v⁰v⁰⁰v⁰⁰⁰

+ m

2P

 v⁰

Z x L

 ∂²

∂t² Z x

0

v⁰²dx

 dx

0!

= mabP (cosΩt), (115)

where

P = ∂

∂t+ τ_ε, (116)

Q = E_R(∂

∂t+ τ_σ). (117)

The symbol, ER, denotes the relaxation Young’s modulus of the beam. It is noted that τε and τσ

are in units of inverse of time.

Let y(x, t) = a(t)φ(x) and take the weak form of the equation of motion by integrating it with respect to its spatial coordinate. Then, the following Duffing-like equation of motion is obtained.

d²a

dτ² + 2βda

dτ + ω²₀a + [α1a³+ α2(a ˙a²+ a²¨a)] = f cosΩτ. (118)

After suitable renormalization, Eq (118) can be expressed as follows.

d²a

dt² + a + 



α1a³+ α2 a ˙a²+ a²a
+ η¨ da

dt − γa − F cost



= 0. (119)

Here

α = α₁

ω₀², (120)

η = 2β

ω₀, (121)

F = f

ω₀², (122)

Ω = ω0

p1 − γ. (123)

The solution of Eq. (119) can be obtained rigorously from the perturbation method [10], as follows.

a = a₀+ a₁+ ²a₂+ · · · , (124) t = T₀+ T₁+ ²T₂+ · · · . (125) Applying the expansion to Eq. (119) with α2 = 0, one obtains the following solution.

a0 = r (T1) cos (T0+ Φ (T1)) , (126) r⁰ = −1

2(ηr + F sinΦ) , (127)

Φ⁰ = −1 8



4γ − 3αr²+4F r cosΦ



. (128)

The steady-state solution of Eqs. (127) and (128) gives rise to the amplitude-frequency equation as follows.

r²

"

η²+ 3

4α1r²− γ

2#

= F². (129)

6 Results and discussion

Comparisons between Abaqus finite element calculation and classical elastica theory have been performed. The results of horizontal displacement are shown in Figure 2, and those of the vertical displacement are in Figure 3. It can be seen that linear solutions from the assumption of small displacement are not valid for horizontal displacement, but valid for small portion of the vertical displacement.

Our finite element calculations are generally in agreement with the classical theory of elastica. In our mathematical derivation above, we have further shown that the theory of elastica is consistent with Euler’s buckling theorem. In Figures 2 and FigV-disp, it can be seen that the prediction from theory of elastica is smaller than that of finite element. The reason is because the elastica theory neglects shear and axial deformation. However, by naive superposition of the contribution of shear

deformation into the results of elastica calculations, one finds it can bring the two predictions closer.

However, it is generally wrong by linearly superimposing a contribution into a nonlinear solution.

We have also tested the effects of number of elements that are used in the finite element technique.

It is found that when the number of elements is moderately enough, increasing the number of elements does provide significant improvements on the solutions. However, there will be always some differences between the finite element solution and theory of elastica due to the in-extensibility assumption in the theory of elastica.

Figure 4 shows a typical result of resonant ultrasound spectroscopy. In this case, the sample was Sn in the shape of 7 mm cube. The peak corresponds to resonance in shear mode. It is estimated that shear modulus of the specimen was about 19 GPa, in agreement with literature results.

7 Conclusions

The progress of the project is on schedule. In two and a half months of time since the project started, basic governing equations have been reviewed and derived. Moreover, some solutions of the equations are obtained for further numerical calculation to compare with experimental data. Our finite element results have revealed the drawbacks of theory of elastica, due to lack of consideration of shear and axial deformation. However, it has been shown a general consistency between classical and numerical solutions. Since the funding of the project was cut for some reasons, it is difficult to obtain necessary equipments to fully complete resonant frequency device. Hence, if my group cannot produce experimental data, we are planning to compare our theoretical results to data in the literature.

8 Self-accessment

According to what we have been done in (1) derivation of governing equation for the three- dimensional viscoelastic elastica, (2) solutions of the weakly nonlinear Duffing equation to model the behavior of the elastic, (3) development of resonant ultrasound spectroscopy, and (4) complete initial stage of developing resonant frequency device, we found ourselves on schedule in the project.

However, there are several points that we need to pursue and complete. (1) Numerical calculation of the solution of the Duffing’s equation to fully characterize the nonlinear dynamical behavior of the cantilever beam. (2) Correlation between theoretical prediction and experimental data. (3) Inverse calculation from experimental data to identify the material’s properties of the vibrating cantilever beam. (4) Possible control of nonlinear processes of the vibrating cantilever. Further experimental work on thin film materials via resonant ultrasound spectroscopy is under way.

9 Acknowledgment

The authors are grateful for the NSC grant, under the contract NSC 96-2221-E-006-068.

References

[1] Cusumano, J. P. and Moon, F. C., Chaotic non-planar vibrations of the thin elastica. Part I:

Experimental observation of planar instability, Journal of Sound and Vibration 179, 185-208 (1995)

[2] Cusumano, J. P. and Moon, F. C., Chaotic non-planar vibrations of the thin elastica. Part II: Derivation and analysis of a low-dimensional model, Journal of Sound and Vibration 179, 209-226 (1995)

[3] Crespo da Silva, M. R. M., Flexural-flexural oscillations of Becks column subject to a planar harmonic excitation, Journal of Sound and Vibration 60, 133-144 (1978)

[4] Crespo da Silva, M. R. M., Harmonic non-linear response of Becks column to a lateral excita- tion, International Journal of Solids and Structures 14, 987-997 (1978)

[5] Crespo da Silva, M. R. M. and Hodges, D. H., Nonlinear flexure and torsion of rotating beams, with application to helicopter rotor blades. I: Formulation, Vertica 10, 151-169 (1986)

[6] Crespo da Silva, M. R. M. and Hodges, D. H., Nonlinear flexure and torsion of rotating beams, with application to helicopter rotor blades. II: Response and stability results, Vertica 10, 171- 186 (1986)

[7] Crespo da Silva, M. R. M., Zaretzky, C. L., and Hodges, D. H., Effects of approximations on the static and dynamic response of a cantilever with a tip mass, International Journal of Solids and Structures 27, 565-583 (1991)

[8] Kane, T. R., Ryan, R. R., and Banerjee, A. K., Dynamics of a cantilever beam attached to a moving base, Journal of Guidance 10, 139-151 (1987)

[9] Misra, A., Hirth, J. P. and Kung, H., Single-dislocation-based strengthening mechanicsm in nano-scale metallic multilayers, Philosophical Magazine A 82(16), 2935-2951 (2002)

[10] A. H. Nayfeh. Introduction to Perturbation Techniques, Wiley, New York (1981)

[11] P. Malatkar and A. H. Nayfeh, ”On the transfer of energy between widely spaced modes in structures”, Nonlinear Dynamics, 31, 225-242 (2003)

[12] Pai, P. F. and Nayfeh, A. H., A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects, International Journal of Solids and Structures 31, 1309-1340 (1994)

[13] Can-zhang Xiao, Yi-zhou Ji and Bao-ping Chang, ”General dynamic equation and dynami- cal characteristics of viscoelastic Timoshenko beams”, Applied Mathematics and Mechanics (English Edition), 11(2), 177-184 (1990)

Figure 1: Schematic of the cantilever under tip bending.

0 2 4 6 8 10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

EI PL

Small Displacement L

δ

Large Displacement

2

sin 1 2

PL EI L

b

h θ

δ = −

ABAQUS Linear Analysis 160 Elements ABAQUS Non-Linear Analysis 160 Elements

ABAQUS Non-Linear Analysis 1000 Elements

Max. Value

N.A. Value (Nodal 1106)

Figure 2: Horizontal displacement of an elastica under tip load bending.

0 2 4 6 8 10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

EI PL

Large Displacement

Small Displacement L

δ

)]

, ( ) ( 4 [

1 ₂ φ

δ _{E k E k}

PL EI L

v = − −

EI PL L

v

2

3

=1 δ

Max. Value N.A. Value

(Nodal 1106)

ABAQUS Linear Analysis 160 Elements ABAQUS Non-Linear Analysis 160 Elements

ABAQUS Non-Linear Analysis 1000 Elements Bernoulli-Euler Thm.

(Only Consider Bending Deformation) Combine Timoshenko's Thm.

(Shear Force Derformation)

Figure 3: Vertical displacement of an elastica under tip load bending.

Figure 4: A typical result of resonant ultrasound spectroscopy. The peak indicates a 7 mm Sn cube”s resonant frequency for shear mode. Estimated shear modulus of the peak is about 19 GPa, consistent with literature results.