Acoustoelastic guided waves in waveguides with arbitrary prestress

Journal Pre-proof Acoustoelastic guided waves in waveguides with arbitrary prestress Peng Zuo, Xudong Yu, Zheng Fan

PII:

S0022-460X(19)30676-5

DOI:

https://doi.org/10.1016/j.jsv.2019.115113

Reference:

YJSVI 115113

To appear in:

Journal of Sound and Vibration

Received Date: 11 July 2019 Revised Date:

19 November 2019

Accepted Date: 21 November 2019

Please cite this article as: P. Zuo, X. Yu, Z. Fan, Acoustoelastic guided waves in waveguides with arbitrary prestress, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/ j.jsv.2019.115113. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Acoustoelastic guided waves in waveguides with arbitrary prestress Peng Zuoa , Xudong Yu∗,a,b , Zheng Fan∗,a a School

of Mechanical and Aerospace Engineering, Nanyang Technological University,50 Nanyang Avenue, Singapore 639798, Singapore b School of Astronautics, Beihang University, Beijing 100191, China

Abstract Acoustoelastic guided wave is promising for prestress measurements in waveguides. However, most of the studies are limited to simple waveguides (e.g. plates and rods) with normal stress. In this paper, a method is developed by considering the acoustoelastic effect in the semi-analytical finite element (SAFE) model, and implemented into a commercial software package. It provides a generalized tool to study acoustoelastic guided waves in waveguides with any cross sections under arbitrary prestress conditions. The method is first validated in a plate under two different prestress conditions. Then it is applied to two practical cases, a rectangular bar subjected to a hydrostatic pressure and an aluminum plate under simple shear deformation, to demonstrate the capability of the method. For the first time, the effect of shear stress on guided wave propagation has been discussed. Key words: Acoustoelasticity, Guided waves, Arbitrary prestress

∗ Corresponding

author. Email addresses: [email protected] (Xudong Yu), [email protected] (Zheng Fan)

Preprint submitted to Elsevier

November 17, 2019

1

1. Introduction

2

Nondestructive measurement of prestress (such as applied or residual stress) in materials is important in

3

engineering, as it significantly affects the mechanical performance of a structure [1, 2]. Acoustoelastic effect,

4

which correlates the change in the acoustic wave velocity to a given stress field, provides a good potential to

5

measure prestress non-destructively. In theoretical studies, the acoustoelastic effect can be described by finite

6

nonlinear elasticity theory (also known as small-on-large) [3, 4] and weakly nonlinear elasticity theory [5, 6].

7

The former allows for large deformations and is mostly used to describe the acoustoelastic effect in soft

8

solids, such as gels and tissues, whereas the latter considers a small pre-deformation and is commonly used

9

in studying the acoustoelastic effect in ordinary stiff solids, such as metals. The present study is set within

10

the framework of weakly nonlinear elasticity theory, which focuses on elastic wave propagation in prestressed

11

stiff solids.

12

Even though the study of prestressed solid can be traced to the work from Rayleigh [7], Love [8] and

13

Biot [9], the modern theory of acoustoelasticity was initially derived in 1953 by Hughes and Kelly [5] for

14

bulk wave propagation in a prestressed isotropic material, where they used Murnaghan’s finite deformation

15

theory and considered third order elastic constants. The theory was then extended to the case of materials

16

with arbitrary symmetry to determine third order elastic constants [6, 10]. Afterwards, Pao and Gamer [11]

17

formulated governing equations of acoustoelastic bulk wave in both initial and natural frame to establish

18

a comprehensive understanding of the relationship between bulk wave speed change and stress level. Such

19

understanding led to extensive applications of the acoustoelastic bulk wave in prestress measurement, such

20

as measuring residual stress in welded structures [12] and rail tracks [13], monitoring the tightening force of

21

bolts [14, 15] and assessing stress levels in bars [16].

22

Ultrasonic guided waves provide the potential for rapid screening of large areas, and therefore have

23

attracted significant attention in recent years. Different from ultrasonic bulk waves, there are hundreds of

24

modes existing in the waveguides because of the boundary constrain. Thus, understanding the effect of

25

stress on different types of guided waves has drawn considerable attention over the last decades. Gandhi

26

et al. [17] extended acoustoelastic bulk wave theory to Lamb waves and explored the anisotropic nature of

27

the dispersion curves under an applied biaxial stress field in an isotropic plate. Based on this anisotropic

28

property of Lamb waves, Shi et al. [18] developed a strategy to estimate a biaxial stress field using a spatially

29

distributed array of guided wave sensors. Concerning about the relatively small velocity change induced by

30

stress, Pei and Bond [19, 20] moved to higher order Lamb modes and demonstrated that higher order modes

31

have a significant higher sensitivity to stress. Except for isotropic plates, Kubrusly [21] further studied the

32

acoustoelastic phenomenon on the dispersion spectra of guided waves for a general anisotropic plate and the

33

study indicated that acoustoelastic effect in anisotropic plates can be complex, even the anisotropy is weak.

34

In addition to plate-like structures, a few studies have investigated the acoustoelastic effect in rods [22] and 2

35

some other waveguides with complex shapes, such as wire strands [23] and rail tracks [24–26].

36

Inhomogeneous stress fields are known to exist extensively in structures that originate from external

37

forcing (e.g. bending) and manufacturing processes (e.g. rolling, drawing, forging and welding). As such,

38

studies on guided wave propagation in inhomogeneous prestress fields have been conducted. Husson [27]

39

studied Lamb wave propagation in a nonuniformly stressed plate through perturbation approaches and the

40

researcher concluded that antisymmetric profiles, such as bending stresses, have less effect on Lamb modes.

41

Lematre et al. [28] modeled the propagation of guided waves in piezoelectric structures subjected to a

42

prestress gradient through the sublayer approach where each layer was assumed to be under a homogeneous

43

prestress. Semi-analytical finite element (SAFE) method was also adopted by Peddeti and Santhanam [29] to

44

model the acoustoelastic effect of nonuniform stress profile on guided waves in isotropic plates. In this case,

45

the nonuniform stress profile was numerically simulated. More recently, Dubuc et al. [30] used a spectral

46

method to compute guided modes in nonuniformly stressed plates.

47

However, most of the work on acoustoelastic guided waves in the literature are limited to structures

48

with simple geometry, such as plates and rods, as analytical solutions such as superposition of partial bulk

49

wave (SPBW), are available. Although some attempts have been performed to study complex waveguides,

50

such as rail tracks [24–26, 31], weakly nonlinear constitutive equations are not involved in the modeling,

51

and homemade codes must be developed to compute the equations numerically, which is tedious and time

52

consuming. On the other hand, only normal stresses are considered in the studies. Shear stress, as the

53

other important component of the stress field, may generate a significant rotation, thereby exerting a non-

54

negligible effect on the propagation of guided waves. Therefore, in order to understand the interaction

55

between prestress field and guided wave propagation comprehensively, shear stress effect on guided wave

56

propagation must be determined.

57

The present study aims to develop an acoustoelastic guided wave model by combining the SAFE method

58

and the acoustoelastic effect (SAFE-Prestress method for short) for waveguides with any cross sections

59

subjected to arbitrary prestress conditions. The governing equations of the acoustoelastic guided waves

60

are implemented into a commercial software to provide an easy access to this method. In addition, with

61

consideration of rotation tensor, it is possible to study shear stress acoustoelasticity. The method is first

62

validated by comparing with two examples in the literature: an aluminum plate subjected to a tensile stress

63

which is perpendicular to the wave propagation and a plate subjected to bending deformation. Then, it is

64

applied to two practical cases: one is to study the guided wave in a rectangular bar under a hydrostatic

65

pressure, which is difficult to solve by analytical methods given the complex cross section; the other case

66

is to study the effect of shear stress on the guided wave propagation in plates, where shear stress causes a

67

significant rotation in the material.

3

68

2. Mathematical framework

69

2.1. Governing equations of acoustoelastic waves with inhomogeneous prestress

70

The analysis for acoustoelastic effect is based on the theory of a small (infinitesimal) elastic wave propa-

71

gating in an elastically prestressed body. In mathematics, three configures of the body have been utilized to

72

demonstrate the material state by using the vectors ξ, X, x with components ξα , XI , xi to denote the position

73

of a material particle in the natural, initial and finial states, respectively. The natural state is the initial,

74

un-deformed configuration of the body. The initial state is the configuration which experiences deformation

75

subjected to prestress, and the final state is the deformed state in the presence of elastic wave propagation.

76

All physical variables and material properties in the three states will be denoted by a superscript “0”, “i”

77

and “f ”, respectively. Deformations from natural to initial state, natural to final state and initial to final

78

state can be given by ui (ui = X − ξ), uf (uf = x − ξ) and u (u = x − X), respectively.

79

In mathematical derivation, three assumptions are adopted: (1) the material is characterized by Mur-

80

naghan material model, where both second and third order elastic constants are considered; (2) the pre-

81

deformation is small and the material remains elastic so that the relationship between the initial stress and

82

initial strain can be approximated by Hooke’s law where nonlinear effect is ignored; (3) the amplitude of

83

the elastic wave propagation in the material is much smaller than the pre-deformation.

84

85

The governing equation of the acoustoelastic wave in the waveguide domain with respect to the natural state is given by [11, 17]   ∂ ∂uγ ∂ 2 uα i ∂uα , Tγβ + Γαβγδ = ρ0 ∂ξβ ∂ξγ ∂ξδ ∂t2

(1)

86

i where ρ0 is the density; u is the incremental displacement from the initial state to the finial state; Tγβ

87

demonstrates the component of the second Piola-Kirchhoff (P-K) stress tensor measured from natural state.

88

A summation over repeated indices is implied here and in the subsequent equations. Γαβγδ = Cαβγδ + Cαβρδ

∂uiγ ∂ui + Cρβγδ α + Cαβγδεη eiεη , ∂ξρ ∂ξρ

(2)

89

where Cαβγδ and Cαβγδεη represent the second and third order elastic constants, respectively; uiα demon-

90

strates the initial displacement from the natural state to the initial state and eiεη is the component of the

91

initial Cauchy strain tensor.

92

Considering that the initial stress or initial strain is inhomogeneous, Eq. (1) can be written as i Tδβ δαγ + Γαβγδ


∂ 2 uγ
∂uγ ∂ ∂ 2 uα i + Tδβ δαγ + Γαβγδ = ρ0 . ∂ξδ ∂ξβ ∂ξβ ∂ξδ ∂t2

(3)

93

As the pre-deformation is assumed to be small where the initial stress and initial strain can be related

94

approximately by Hooke’s law in its most general form, the second P-K stress for the initial state can be

95

written as i Tδβ = Cδβλρ eiλρ . 4

(4)

96

With mathematical manipulation, Eq. (3) can be simplified as Aαβγδ

∂ 2 uγ ∂uγ ∂ 2 uα , + Dαγδ = ρ0 ∂ξδ ∂ξβ ∂ξδ ∂t2 ∂ui

97

98

99

100

101

102

103

105

106

∂(Aαβγδ ) . ∂ξβ

In small

deformation theory, displacement gradient can be demonstrated through decomposing it into its symmetric  i   i  ∂u ∂ui ∂u ∂ui i (Cauchy strain tensor, eiγρ = 21 ∂ξργ + ∂ξγρ ) and antisymmetric (rotation tensor, rγρ = 21 ∂ξργ − ∂ξγρ ) ∂uiγ ∂ξρ

i = eiγρ + rγρ . Therefore, the coefficient Aαβγδ can be rewritten as Aαβγδ = Cδβλρ eiλρ δαγ +

i i Cαβγδ + Cαβρδ eiγρ + rγρ + Cρβγδ eiαρ + rαρ + Cαβγδεη eiεη .

parts by

On the boundary, the incremental stress tensor is adopted to derive the incremental stress boundary condition by [17, 32] τα = Bαβγδ

104

∂ui

where Aαβγδ = Cδβλρ eiλρ δαγ + Cαβγδ + Cαβρδ ∂ξργ + Cρβγδ ∂ξρα + Cαβγδεη eiεη , Dαγδ =

(5)

∂uγ nβ , ∂ξδ

(6)

where τα is the component of the incremental traction vector at the boundaries associated with wave
i + Cαβγδεη eiεη and nβ is the outward unit vector in the propagation; Bαβγδ = Cαβγδ + Cαβρδ eiγρ + rγρ natural state.

107

From Eqs. (5) and (6), it can be seen that the governing equations of acoustoelasticity are analogous to

108

that of the classical elastodynamic problem for the elastic body free of prestress, except that the effect of the

109

prestress is completely integrated into the coefficients Aαβγδ , Dαγδ and Bαβγδ , leading to the coefficients

110

being the function of the initial strain and initial rotation. If the prestress is zero, the coefficients are

111

degenerated to the classical form. Thus, the initial strain and initial rotation have to be determined before

112

solving the governing equations of acoustoelastic guided wave.

113

2.2. Semi-analytical finite element method

114

SAFE method is known to be powerful and widespreadly used for modal study in waveguides with

115

arbitrary cross sections. It starts with the three dimensional elasticity approach and reduces the three

116

dimensional problem into a two dimension one by using an assumption of a harmonic guided wave propa-

117

gation along the axial direction, leading to a significant reduction on computation cost [33]. In addition,

118

the governing equations of SAFE method can be transformed and implemented into a commercial software

119

package [34], providing an easy access to modal studies of waveguides [35–38].

120

In this part, the governing equations of acoustoelastic waves (Eqs. (5) and (6)) are simplified by using

121

the SAFE method and implemented into the commercial software package providing an easy access. Fig. 1

122

plots a waveguide with arbitrary cross section in the natural state.

123

124

Assuming a guided wave propagating in the ξ3 direction, the displacement in the waveguide can be described by uα (ξ1 , ξ2 , ξ3 , t) = Uα (ξ1 , ξ2 ) eI(kξ3 −ωt) , 5

(7)

Figure 1: Schematic of an arbitrary cross section of a waveguide; the waveguide is infinite in the ξ3 direction; Ω and ∂Ω represent the waveguide domain and waveguide boundary, respectively.

125

in which Uα demonstrate the displacement in the cross section; I and k represent the imaginary unit and

126

wavenumber of the guided wave in the ξ3 direction, respectively; ω = 2πf is the angular frequency with f

127

being the frequency; the subscript α = 1, 2, 3. Thus the derivative of the displacement can be written as   α α I(kξ3 −ωt)  ∂u = ∂U  ∂ξ1 e  ∂ξ1 ∂uα ∂Uα I(kξ3 −ωt) . (8) ∂ξ2 = ∂ξ2 e     ∂uα = IkU eI(kξ3 −ωt) ∂ξ3

α

128

Substituting Eqs. (7) and (8) into Eq. (5), the governing equations in the waveguide domain can be cast

129

into   3 X ∂uγ ∂ 2 uγ + Dαγδ + ω 2 ρ0 uα = 0 for α = 1, 2, 3. Aαβγδ ∂ξδ ∂ξβ ∂ξδ

(9)

β,γ,δ=1 130

Using some intermediary transformations, Eq. (9) can be written as ∂2U

Aαβγδ ∂ξδ ∂ξγβ + I (Aα3γδ + Aαδγ3 )

∂(kUγ ) ∂ξδ

− kAα3γ3 (kUγ ) + Dαγδ

+IDαγ3 (kUγ ) + ω 2 ρ0 Uγ δαγ = 0 131

with summation over the indices γ = 1, 2, 3 and β, δ = 1, 2.

6

∂Uγ ∂ξδ

,

(10)

132

On the boundary (∂Ω), the incremental stress boundary condition is written as 3 P

τα =

∂u

Bαβγδ ∂ξδγ nβ

β,γ,δ=1 " 3 2 P P

=

γ=1 133

β,δ=1

∂u Bαβγδ ∂ξδγ nβ

+

2 P β=1

∂u Bαβγ3 ∂ξ3γ nβ

δ=1

# . ∂u Bα3γδ ∂ξδγ n3

+

(11)

∂u Bα3γ3 ∂ξ3γ n3

Considering the fact that n3 = 0 on the surface of the waveguide, then   3 2 2 X X X ∂u γ  τα = nβ + Bαβγ3 Ikuγ nβ . Bαβγδ ∂ξ δ γ=1

(12)

β=1

β,δ=1

134

+

2 P

Excluding the phase term eI(kξ3 −ωt) , Eq. (12) can be cast into Sα = Bαβγδ

∂Uγ nβ + IBαβγ3 (kUr ) nβ , ∂ξδ

(13)

135

with summation over the indices γ = 1, 2, 3 and β, δ = 1, 2, where Sα is the mode shape of τα at the

136

boundaries.

137

138

Equation (10) represents a quadratic eigenvalue problem. In order to cast it into a general linear form, a new variable should be introduced Vγ = kUγ .

139

(14)

Thus the governing equations of the acoustoelastic guided wave in the waveguide can be rewritten as ∂2U

Aαβγδ ∂ξδ ∂ξγβ + I (Aα3γδ + Aαδγ3 )

∂Vγ ∂ξδ

− kAα3γ3 Vγ + Dαγδ

∂Uγ ∂ξδ

+IDαγ3 Vγ + ω 2 ρ0 Uγ δαγ = 0 in Ω

,

(15)

140

ρ0 ω 2 δαγ Vγ − kρ0 ω 2 δαγ Uγ = 0 in Ω,

(16)

141

Sα = Bαβγδ 142

143

144

∂Uγ nβ + IBαβγ3 Vγ nβ on ∂Ω, ∂ξδ

(17)

with summation over the indices γ = 1, 2, 3 and β, δ = 1, 2. In a commercial finite element method (FEM) package (COMSOL Multiphysics), the input formula for eigenvalue problems has the general expression as [34] ∇ · (c∇u + αu − r) − au − β · ∇u + da λu − ea λ2 u = f in Ω,

(18)

145

in which u represents the set of variables to be determined; the various coefficients c, α, r, a, β, da , ea , f

146

from the input formula do not have any particular meaning except that they represent functions of the pa-

147

rameters of the problem investigated. The generalized Neumann boundary condition and Dirichlet boundary

148

condition on ∂Ω are expressed in the same FEM code as n · (c∇u + αu − r) + qu = g on ∂ Ω, 7

(19)

149

u = h on ∂Ω,

(20)

150

where q, g and h represents coefficients in the boundaries. In order to cast the governing equations into the

151

general FEM formula, the variables in the FEM software has to be T

u = [U1 , U2 , U3 , V1 , V2 , V3 ] . 152

153

154

155

(21)

And the FEM coefficients must be        cU U 0 0 αU V 0 aU U 0 ; α =  ; r =  ; a =  c= 0 0 0 0 0 0 aV V        0 βU V 0 dU V 0 0 0  ; da =   ; ea =  ; f =  β= 0 0 dV U 0 0 0 0

 ;  .

(22)



0 represents a zero matrix of appropriate dimension and the submatrices are given in Appendix A. It should be noted that, substituting the FEM coefficients into the Neumann boundary conditions, the Neumann boundary conditions are written as Aαβγδ

∂Uγ nβ + IAαβγ3 Vγ nβ = gα on ∂Ω, ∂ξδ

(23)

156

where gα is the coefficient in the boundaries to be determined. To cast the Neumann boundary conditions

157

into the incremental stress boundary conditions, the coefficient gα should be ∂Ur nβ + IFαβγ3 Vr nβ , (24) ∂ξδ
i . In the software package, the derivative of = Cδβλρ eiλρ δαγ + Cρβγδ eiαρ + rαρ gα = Fαβγδ

158

with Fαβγδ = Aαβγδ − Bαβγδ

159

the variables and the outward unit vector can be called and written into the coefficients of the expression

160

directly in the calculation.

161

It can be seen that the Neumann boundary conditions with g =

h

g1

g2

g3

0

0

0

iT

and q = 0

162

correspond to incremental traction free boundary conditions in the waveguides. Dirichlet boundary con-

163

ditions, u = h can be used to implement the incremental displacement boundary conditions on the corre-

164

sponding boundaries. T

165

The solution of the eigenvalue problem is the eigenvalue k and the eigenvector u = [U1 , U2 , U3 , V1 , V2 , V3 ]

166

for chosen values of angular frequency ω and each solution will reveal the wavenumbers of all of the possible

167

modes at that frequency. It is known that there are three types of modes existing in the elastic waveguide

168

with different type of wavenumber: propagating modes (real wavenumber), non-propagating modes (pure

169

imaginary wavenumber) and evanescent modes (complex wavenumber). Propagating modes are of interest,

170

thus they are identified and collected at each solution by selecting the real wavenumber modes. The full 8

171

dispersion curve spectrum is plotted by repeating the eigenvalue solutions over the frequencies of interest,

172

calculating the overlap of the mode shapes at adjacent frequencies which represents the similarity of the

173

mode shape and combining modes with the most similar mode shapes after each frequency step.

174

2.3. Energy velocity

175

In post-processing, phase velocity and group velocity of the propagating modes can be calculated by the ω k

and cg =

dω dk ,

176

definition cp =

respectively. However, the group velocity calculation based on the definition

177

usually requires a numerical process of computing derivatives of the phase velocity dispersion curves. One

178

alternative is to calculate group velocity based on a linear algebra using stiffness matrices, mass matrices

179

and eigenvectors [39]. Another method is to calculate energy velocity, which is identical to the group velocity

180

when the guide wave experiences no dissipation [39]. Thus the group velocity in the ξα direction can be

181

calculated by the following equation [40] R Pξ dΩ cg = ce = RS α , E dΩ S t

(25)

182

where ce is the energy velocity; S is the cross section of the whole geometry; Pξα is the mean energy flux in

183

the ξα direction and Et is the mean energy density which can be calculated by a sum of the mean kinetic

184

energy density and the mean strain energy density (Et = Ek +Ep ). As there is no-damping in the waveguide,

185

the mean strain energy density is equal to the mean kinetic energy density, thus Et = 2Ek is used in the

186

calculation. The mean energy flux and the mean kinetic energy density are defined by Pξα = −hvβ τβα it ,

(26)

1 0 ρ hvα vα it , 2

(27)

187

Ek = 188

where h·it is the time average over one period; vα = −Iωuα demonstrates the velocity component; ταβ is the

189

effective stress tensor in the prestressed body, calculated by the formula ταβ = Aαβγδ uγ,δ [30, 32]. Guided

190

waves are assumed to propagate in the ξ3 direction in the waveguide, thus evaluating the time average, the

191

mean energy flux in the ξ3 direction and mean kinetic energy density can be determined explicitly as 1 Pξ3 = − Re (T13 V1 ∗ + T23 V2 ∗ + T33 V3 ∗ ) , 2

(28)

192

Ek =

1 0 ρ Re (V1 V1∗ + V2 V2∗ + V3 V3∗ ) , 4

(29)

193

where the asterisk stands for the complex conjugate and Re means “the real part of”. The component of

194

the stress tensor in the prestressed body can be calculated by ∂U1 1 T13 = A1311 ∂U ∂ξ1 + A1312 ∂ξ2 + A1313 IkU1 ∂U2 2 , + A1321 ∂U ∂ξ1 + A1322 ∂ξ2 + A1323 IkU2 ∂U3 3 + A1331 ∂U ∂ξ1 + A1332 ∂ξ2 + A1333 IkU3 9

(30)

195

∂U1 1 T23 = A2311 ∂U ∂ξ1 + A2312 ∂ξ2 + A2313 IkU1 ∂U2 2 , + A2321 ∂U ∂ξ1 + A2322 ∂ξ2 + A2323 IkU2

(31)

∂U3 3 + A2331 ∂U ∂ξ1 + A2332 ∂ξ1 + A2333 IkU3 196

∂U1 1 T33 = A3311 ∂U ∂ξ1 + A3312 ∂ξ2 + A3313 IkU1 ∂U2 2 , + A3321 ∂U ∂ξ1 + A3322 ∂ξ2 + A3323 IkU2

+ 197

3 A3331 ∂U ∂ξ1

+

3 A3332 ∂U ∂ξ2

(32)

+ A3333 IkU3

and the velocity components are calculated by V1 = −IωU1 , V2 = −IωU2 , V3 = −IωU3 .

(33)

198

In the post-processing, all of these quantities can be easily extracted from the eigensolutions of the FEM

199

solver.

200

3. Validation of SAFE-Prestress method

201

In this section, a plate with two different types of prestress condition is chosen to validate the developed

202

SAFE-Prestress method. The results from the method are compared with the one in the literature by SPBW

203

method [17, 19] and other finite element methods [29].

204

An isotropic aluminum plate with thickness of 1 mm is considered. In order to simulate wave propagating

205

in an infinitely wide plate, a narrow strip of the plate with width of 0.2 mm is used and coupled with a

206

207

periodic boundary condition (PBC) which represents continuity of displacements and stresses between the h iT two edges, as shown in Fig. 2. Neumann boundary conditions with g = g1 g2 g3 0 0 0 and

208

q = 0 are imposed on the top and bottom surfaces of the plate. The material properties of aluminum

209

6061-T6 given in Table 1. In the simulation, the whole cross section is automatically meshed by the software

210

with 5034 triangular elements with an approximation side length of 0.01 mm, and the degrees of freedom

211

are 61854. Table 1: Material properties used in the calculations, after [17] aluminum, [41] steel, for the third order elastic constants.

Material

ρ0 kg/m3




λ (GPa)

µ (GPa)

l (GPa)

m (GPa)

n (GPa)

Aluminum

2704

54.308

27.174

-281.5

-339

-416

Steel

7932

107.8

84.7

-1110

-472.5

-325

10

Figure 2: Schematic of an aluminum plate coupled with periodic boundary conditions for acoustoelastic guided wave calculations.

11

212

3.1. Guided waves in a plate with a tensile stress

213

The first case is to study guided wave modes in the plate when it is subjected to a 100 MPa uniform

214

tensile stress in the ξ1 direction perpendicular to the wave propagating direction (the ξ3 direction). As the

215

i pre-deformation is caused by the uniform tensile stress, initial rotation becomes zero, rαβ = 0. The Cauchy

216

217

strain tensor can be obtained through inverting Eq. (4) with the initial stress tensor expressed as   T11 0 0     Ti =  0 0 0  ,   0 0 0

(34)

where T11 = 108 .

218

Figure 3 shows velocity change for the low order Lamb waves (including S0, A0, S1 and A1) in the

219

plate caused by the applied stress, where the change in velocity is calculated by computing the difference

220

between the velocity when the plate is subjected to the applied stress and the velocity when the plate is free

221

of the applied stress. Fig. 3(a) and (b) plot the changes in phase velocity and group velocity (equivalent

222

to energy velocity in this case) respectively and the results are compared to the solutions from the SPBW

223

method [17, 19]. It can be seen that the results from the SAFE-Prestress method developed in this paper

224

agree well with the solutions from SPBW method. It should be noted that, all modes can be calculated by

225

our method, including shear horizontal (SH) modes and high order Lamb modes. However, only four low

226

order Lamb modes are shown here for the purpose of comparison.

227

Some interesting features can be obtained from the results. For the S0 mode, it experiences a relatively

228

large velocity change around 2500 kHz. For the higher order Lamb modes, e.g. S1 and A1, both of them

229

have a significant change in velocity when they are close to the cut-off frequencies. It can also be seen that

230

all modes have a relatively large change in velocity at the low frequency region. However these changes in

231

velocity will converge to a constant value when they go to the high frequency region.

232

3.2. Guided waves in a plate with a bending deformation

233

The second validation case is to simulate guided wave modes in a plate subjected to a pure bending

234

deformation. The pure bending deformation is in the ξ2 − ξ3 plane and leads to a normal stress profile

235

which varies linearly along the plate thickness with a peak bending tensile stress and bending compressive

236

stress of 200 MPa on the top and bottom surface of the plate, respectively. The plate bending effect on

237

guide wave propagation has been calculated by sublayer method [28] and finite element method [29], and

238

their results are used for comparison. As the bending normal stress cause very small rotation in the plate,

239

i it is neglected in the computation with rαβ = 0. The Cauchy strain tensor in the coefficient (Aαβγδ ) can be

12

Figure 3: (Color online) Changes in (a) phase velocity and (b) group velocity for the S0, A0, S1 and A1 modes in the plate with an applied tensile stress perpendicular to the wave propagation. The solid lines represent the results from the SAFE-Prestress method; the dots are results from the SPBW method. [17, 19]

240

obtained through inverting Eq. (4) with the initial  0   i T = 0  0

bending stress tensor expressed as  0 0   0 0 ,  0 T33

(35)

241

where T33 is a linear function of the plate thickness expressed as T33 = A ξh2 , with A = 4 × 108 , h = 10−3 (the

242

thickness of the plate used in the modeling). This expression represents a linear variation stress function

243

with a peak value of 200 MPa on the surface of the plate.

244

Figure 4(a) and (b) plot the changes in phase velocity and group velocity respectively for lower order

245

Lamb modes caused by the pure bending deformation and compared to the solutions obtained from other

246

finite element method [29]. Very good agreements can be seen between the two results.

247

It is interesting to find that under the pure bending deformation, the phase velocity of the S0 mode

248

increases with the frequency, while the A0 mode is in an opposite trend simultaneously. This contrasting

249

property is mainly caused by the different nature of the Lamb modes (the S0 mode is symmetric and the A0

250

mode is antisymmetric) and their interactions with the antisymmetric bending stress. For the higher order

251

Lamb modes (S1 and A1), as the frequency increases, the bending stress has a much smaller effect on the

252

guided wave propagation in the plate.

253

4. Applications of SAFE-Prestress method

254

255

In this section, the SAFE-Prestress method is applied to two examples to demonstrate the capability of the method. One example is to study wave propagating in a steel rectangular bar when the bar is subjected 13

Figure 4: (Color online) Changes in (a) phase velocity and (b) group velocity for the S0, A0, S1 and A1 modes in the plate under pure bending deformation. The solid lines represent the results from the SAFE-Prestress method; the dots are results from other finite element method. [29]

256

to a hydrostatic pressure. This case is considered because rectangular bars are typical structures with a

257

wide range of applications in industries, and therefore the study of acoustoelastic effect in rectangular bars

258

is practically important. In addition, because of the rectangular cross section, no analytical method can be

259

used to solve the problem. The other example is to understand wave propagation in a plate under a simple

260

shear deformation, where rotation induced by shear stress is comparable to Cauchy strain tensor, so that it

261

cannot be neglected.

262

4.1. Guided waves in a rectangular bar subjected to a hydrostatic pressure

263

264

265

A steel rectangular bar is considered with the geometry of the cross section being 10 mm × 2 mm, as shown in Fig. 5. A hydrostatic pressure at the level of 200 MPa is applied. Neumann boundary conditions h iT with g = g1 g2 g3 0 0 0 and q = 0 are applied to all boundaries of the cross section and

266

material properties of steel are given in Table 1. In order to determine the coefficients (Aαβγδ ), the Cauchy

267

strain tensor and rotation tensor must be calculated. Because the hydrostatic pressure is normal stresses,

268

i the rotation should be zero, rαβ = 0. The Cauchy strain tensor can be obtained through inverting Eq. (4)

269

with the initial stress tensor expressed as 

T11

  Ti =  0  0 270

271

0 T22 0

0



  0 ,  T33

(36)

where T11 = T22 = T33 = −2 × 108 . In calculation, the whole cross section is meshed by 5036 triangular elements with element size of 0.1 mm and the degrees of freedom are 61878. 14

Figure 5: Schematic of a steel rectangular bar for acoustoelastic guided wave calculations.

272

It is known that the modal properties of rectangular bar free of prestress have been understood well [42,

273

43]. Because of the twofold of symmetry of the rectangular cross section, the modes of wave propagation

274

in the rectangular bar can be classified into three categories: longitudinal modes, torsional modes and

275

bending modes. Longitudinal modes are symmetrical to both ξ1 and ξ2 axis, while the torsional modes are

276

asymmetrical to the ξ1 and ξ2 axis. Bending modes are symmetric to either ξ1 or ξ2 axis, and asymmetric

277

about the other axis. Thus the bending modes can be named as Bξn2 and Bξn1 where the subscript represents

278

the axis of asymmetry and the superscript n is the mode order. Although the prestress may distort the

279

mode shape of the guided wave, leading to the mode shape slightly deviating from the convention one defined

280

above, the naming convention is still adopted in this paper considering the fact that the modes can retain

281

a predominant character.

282

Figure 6 plots the changes and relative changes of both phase velocity and group velocity as a function

283

of frequency for the four fundamental modes when the bar is under a 200 MPa hydrostatic pressure (all

284

modes can be calculated, but only four fundamental modes are shown for conciseness). The relative change

285

of velocity is a dimensionless value which is equal to the change of velocity divided by the velocity of the

286

mode when the bar is free of applied stress. The relative change of velocity indicates the sensitivity of the

287

mode to the applied stress. Fig. 6(a) and (b) show the changes and relative changes of the phase velocity.

288

It can be seen that all modes have relatively large phase velocity changes at the low frequency region (less

289

than 200 kHz). As frequency increases, the changes of phase velocities for some modes merge into constant

290

values (the L1 and By1 modes merge into a value of 4 m/s, while the T1 and Bx1 modes merge into a value

291

of 3.1 m/s). For the L1 mode, even though it has a large phase velocity change at the low frequency (about

292

13 m/s), the relative change of the phase velocity is only about 0.25% according to Fig. 6(b). For the Bx1

293

mode and the T1 mode, it also can be noticed that, their velocities have quite large variation at the low

294

frequency region (less than 100 kHz), which may provide an attractive zone for prestress measurement.

295

For changes in group velocity (Fig. 6(c) and (d)), similar phenomena can be identified: the variations

296

of group velocities are large at the low frequency region and they merge to some constants at the high

297

frequency region. It is interesting to point out that the L1 mode has a maximum value on group velocity 15

Figure 6: (Color online) Changes and relative changes in ((a) and (b)) phase velocity and ((c) and (d)) group velocity for the four fundamental modes in the rectangular bar subjected to a 200 MPa hydrostatic pressure.

298

change around 230 kHz, which may lead to good sensitivity to the applied stress at this frequency.

299

4.2. Guided waves in an aluminum plate under simple shear deformation

300

This case is to study the effect of initial shear deformation on the guided wave propagation in an

301

aluminum plate, as shown in Fig. 7, where the thickness and width of the plate are assumed to be 1 mm and

302

0.2 mm respectively with PBCs being applied to the left and right side of the plate to simulate continuity

303

of displacements and stresses. The shear deformation is set in the ξ1 − ξ2 plane. Material properties are

304

305

306

307

obtained from Table 1. On the top and bottom surface of the plate, Neumann boundary conditions with h iT g = g1 g2 g3 0 0 0 and q = 0 are implemented. In order to generate rotation in the plate, a simple shear deformation is applied with the initial displacement being assumed to be [44] uiξ1 = A ξh2 , uiξ2 = 0 and uiξ3 = 0, in which A is a constant representing the shear deformation extent, h = 10−3 is the 16

Figure 7: Schematic of a plate subjected to a simple shear deformation for acoustoelastic guided wave calculations.

308

thickness of the plate. In this study, A = 4 × 10−6 is adopted to generate a shear stress linearly varying

309

with the thickness of the plate with a maximum value of approximate 100 MPa on the surface of the plate.

310

In order to determine the coefficients (Aαβγδ ), Cauchy strain tensor and rotation tensor can be calculated

311

according to the initial displacement field,    ei =  

0

A 2h

A 2h

0

0

0

0

A 2h

0



  0 ,  0

(37)

312



  A ri =  − 2h  0

0 0

0



  0 .  0

(38)

313

The whole cross section is meshed with 5034 triangular elements with element length of 0.01 mm, and the

314

degrees of freedom are 61854.

315

Figure 8 plots the phase velocity dispersion curves for the Lamb modes (S0, A0, S1, A1) and SH modes 17

Figure 8: (Color online) Phase velocity dispersion curves for the Lamb modes (represented by solid lines) and SH modes (represented by dash lines) in an aluminum plate when the plate is under a simple shear deformation.

316

(SH0, SH1, SH2, SH3) when the plate is under the initial shear deformation. It is interesting to find out

317

that strong mode coupling occurs between the SH modes and the Lamb modes at specific frequency regions

318

when their phase velocity dispersion curves get close to each other, which have been marked with dashed

319

boxes in the figure: they are SH1 mode coupling with the S0 mode around 2300 kHz, SH2 mode being

320

coupled with the A1 mode around 4500 kHz and the SH3 mode having a coupling effect with the S1 mode

321

in a relatively large range from 6000 kHz to 9000 kHz.

322

The mode coupling causes the phase velocity dispersion curves repel each other at the frequencies, leading

323

to a significant change in phase velocity and group velocity. Fig. 9(a) and (b) show changes and relative

324

changes in phase velocities for the S0, A0, SH0 and SH1 mode. It can be seen that the phase velocities of

325

the coupled modes (SH1 and S0) have abrupt changes within a very narrow frequency region (from 2000

326

kHz to 3000 kHz). For the SH1 mode, the change in phase velocity increases dramatically to a maximum

327

value (about 30 m/s) and then decreases sharply to a value close to 0. For the S0 mode, the change in phase

328

velocity has an opposite trend: it decreases first and then increases to a value close to 0. It also can be seen

329

that when the change in phase velocity approaches the maximum value, the relative change can approach

330

approximately 0.7%. Fig. 9(c) and (d) plot the changes and relative changes in phase velocity for the S1,

331

A1, SH2 and SH3 mode. A similar phenomenon for the coupling modes (SH2 and A1; SH3 and S1) can be

332

seen except that the coupling frequency region of higher order modes is much larger than that of the SH1 18

Figure 9: (Color online) Changes and relative changes in phase velocity for ((a) and (b)) S0, A0, SH0 and SH1 mode; and for ((c) and (d)) S1, A1, SH2 and SH3 mode in the plate under a simple shear deformation.

333

mode and S0 mode.

334

The changes and relative changes in group velocity are shown in Fig. 10. Being different from the changes

335

in phase velocity, the changes in group velocity for the SH modes increase to a maximum value and then

336

decrease to a negative value and finally increase to a value close to zero. The changes in group velocity for

337

the Lamb modes have an opposite trend. Furthermore, it is interesting to find out from Fig. 10(b) and (d)

338

that the relative change in the group velocity can reach to as much as 5%, indicating excellent sensitivity

339

to shear stress at certain frequencies, which may provide a great potential in non-destructive evaluation of

340

shear stress.

341

In order to further demonstrate the effect of mode coupling, Fig. 11 shows dispersion curves for the

342

S0 mode and the SH1 mode with mode shapes of these two modes at different frequencies. It has been

343

understood that the S0 mode is symmetric and the SH1 mode is antisymmetric when the plate is free of 19

Figure 10: (Color online) Changes and relative changes in group velocity for ((a) and (b)) S0, A0, SH0 and SH1 mode; and for ((c) and (d)) S1, A1, SH2 and SH3 mode in the plate under a simple shear deformation.

20

Figure 11: (Color online) Phase velocity dispersion curves for the S0 mode and the SH1 mode in a plate under initial shear deformation with insets showing the mode shape of the S0 mode at 1000 kHz, 2300 kHz and 3500 kHz, and the mode shape of the SH1 mode at 2000 kHz, 2300 kHz and 3500 kHz, where the color indicates the mean energy flux in the ξ3 direction.

344

prestress. However, the initial shear deformation change their properties. From Fig. 11, it is obvious that,

345

both the SH1 mode and the S0 mode keep their symmetry properties at frequencies lower than the coupling

346

frequencies. As frequency increases, when they go to the model coupling frequency region, both of them

347

are distorted, with the S0 mode becoming antisymmetric. However, when they go to the higher frequency

348

region, away from the coupling region, there is almost no coupling effect between these two modes. It should

349

be noted that, in anisotropic plates, the mode coupling is usually caused by material anisotropic properties

350

in nonsymmetry directions [21, 45]. In this case, there is no material anisotropy in the plate, thus the mode

351

coupling is due to the prestress induced anisotropy.

21

352

5. Conclusions

353

In this work, a method has been developed by combining the acoustoelastic effect with the SAFE method,

354

and implemented into a commercial software package. It provides an easily accessible approach for studying

355

acoustoelastic guided wave in waveguides with any cross sections subjected to arbitrary prestress. The

356

method has been validated on aluminum plate when the plate is under both a tensile stress perpendicular to

357

the wave propagation direction and a pure bending deformation, showing perfect agreement with solutions

358

with other methods. The method is then applied to two practical examples. The first case is to study guided

359

waves in a rectangular bar subjected to a hydrostatic pressure to demonstrate the capability of the method.

360

It is found that the velocity change for the four fundamental modes is relatively large at the low frequency

361

region, and merges to some constant values at the high frequency region. The second case is to study the

362

effect of shear stress on guided wave propagation, where shear deformation can cause a significant rotation in

363

the waveguide. In this case, strong mode coupling between SH modes and Lamb modes at specific frequency

364

regions can be found when their phase velocity dispersion curves get close to each other. The mode coupling

365

causes a significant change in both phase velocity and group velocity, leading to a potential non-destructive

366

measurement for shear stress. Although the material used in the application cases in this paper are isotropic,

367

the method can be easily extended to anisotropic materials, where the material anisotropy and the prestress

368

induced anisotropy may significantly complicate the studies on acoustoelastic guided waves.

369

It should be noted that, in the current work, elastic pre-deformation is assumed, which leads to a Hooke’s

370

relation between the initial stress and the initial strain. However, residual stress in bodies usually arises

371

from processes that are not elastic. Thus the relationship between the residual stress and the guided wave

372

velocity change developed by the SAFE-Prestress method may be compromised in practical measurements

373

where plastic deformation occurs.

374

Acknowledgements

375

376

377

This work was supported by MOE AcRF Tier 1, RG99/17.

A. The coefficients of the FEM formula According to Eq. (22), the coefficients in the FEM formula are given as follows.   U UU UU c c cU 12 13   11  U UU UU  cU U =  cU c22 c23  ,   21 U UU UU cU c c 31 32 33

22

(A.1)

378

U with cU αβγδ = Aαγβδ , then

 U  cU 11 =

U cU 1111

U cU 1112

U cU 1121

U cU 1122





=

A1111

A1112

A1211

A1212

A1121

A1122

A1221

A1222

A1131

A1132

A1231

A1232

A2111

A2112

A2211

A2212

A2121

A2122

A2221

A2222

A2131

A2132

A2231

A2232

A3111

A3112

A3211

A3212

A3121

A3122

A3221

A3222

A3131

A3132

A3231

A3232

 

(A.2)

379

 U  cU 12 =

U cU 1211

U cU 1212

U cU 1221

U cU 1222

U cU 1311

U cU 1312

U cU 1321

U cU 1322

U cU 2111

U cU 2112

U cU 2121

U cU 2122

U cU 2211

U cU 2212

U cU 2221

U cU 2222

U cU 2311

U cU 2312

U cU 2321

U cU 2322

U cU 3111

U cU 3112

U cU 3121

U cU 3122

U cU 3211

U cU 3212

U cU 3221

U cU 3222

U cU 3311

U cU 3312

U cU 3321

U cU 3322





=

 ;

(A.3)

380

 U  cU 13 =





=

 ;

(A.4)

381

 U  cU 21 =





=

 ;

(A.5)

382

 U  cU 22 =





=

 ;

(A.6)

383

 U  cU 23 =





=

 ;

(A.7)

384

 U  cU 31 =





=

 ;

(A.8)

385

 U  cU 32 =





=

 ;

(A.9)

386

 U  cU 33 =





=

 .

(A.10)

387



UV α11

UV α12

  UV αU V =  α21  UV α31 388

UV α22 UV α32

UV α13



 UV  , α23  UV α33

(A.11)

UV with ααβγ = IAαγβ3 , then

 UV α11 =

UV α111 UV α112





= 23

IA1113 IA1213

 ;

(A.12)

389

 UV α12 =

UV α121 UV α122





=

IA1123 IA1223

 ;

(A.13)

390

 UV α13 =

UV α131 UV α132





=

IA1133 IA1233

 ;

(A.14)

391

 UV α21 =

UV α211 UV α212





=

IA2113 IA2213

 ;

(A.15)

392

 UV α22 =

UV α221 UV α222





=

IA2123 IA2223

 ;

(A.16)

393

 UV α23 =

UV α231 UV α232





=

IA2133 IA2233

 ;

(A.17)

394

 UV α31 =

UV α311 UV α312





=

IA3113 IA3213

 ;

(A.18)

395

 UV α32 =

UV α321 UV α322





=

IA3123 IA3223

 ;

(A.19)

396

 UV α33 =

UV α331 UV α332





=

IA3133 IA3233

 .

(A.20)

397





−ρ0 ω 2

0

0

0

−ρ0 ω 2

0

0

0

0

−ρ ω

−ρ0 ω 2

0

0

0

−ρ0 ω 2

0

0

0

−ρ0 ω 2

  aU U =  

2

  , 

(A.21)

398

   aV V =  

   . 

(A.22)

399



UV β11

  UV β U V =  β21  UV β31

UV β12

UV β13

UV β22

UV β23

UV β32 24

UV β33

   , 

(A.23)

400

UV with βαβγ = −IAα3βγ , then

 UV β11 =

UV β111 UV β112





=

−IA1311 −IA1312

 ;

(A.24)

401

 UV β12 =

UV β121 UV β122





=

−IA1321 −IA1322

 ;

(A.25)

402

 UV β13 =

UV β131 UV β132





=

−IA1331 −IA1332

 ;

(A.26)

403

 UV β21 =

UV β211 UV β212





=

−IA2311 −IA2312

 ;

(A.27)

404

 UV β22 =

UV β221 UV β222





=

−IA2321 −IA2322

 ;

(A.28)

405

 UV β23 =

UV β231 UV β232





=

−IA2331 −IA2332

 ;

(A.29)

406

 UV β31 =

UV β311 UV β312





=

−IA3311 −IA3312

 ;

(A.30)

407

 UV β32 =

UV β321 UV β322





=

−IA3321 −IA3322

 ;

(A.31)

408

 UV β33 =

UV β331 UV β332





=

−IA3331 −IA3332

 .

(A.32)

409



−A1313

  dU V =  −A2313  −A3313

−A1323 −A2323 −A3323

−A1333



  −A2333  ,  −A3333

(A.33)

410

   dV U =  

−ρ0 ω 2

0

0

0

−ρ ω

0

0 25

0 2

0 −ρ0 ω 2

   . 

(A.34)

411

For isotropic material, the second and third order elastic constants can be calculate by Cαβγδ = λδαβ + µ (δαγ δβδ + δαδ δβγ ) ,

(A.35)

412

Cαβγδεη = 2 l − m + +

n 2

n 2




δαβ δγδ δεη + 2 m −

n 2




(δαβ Iγδεη + δγδ Iεηαβ + δεη Iαβγδ )

,

(A.36)

(δαγ Iβδεη + δαδ Iβγεη + δβγ Iαδεη + δβδ Iαγεη )

(δαγ δβδ +δαδ δβγ ) ; 2

413

where Iαβγδ =

λ, µ are Lam´e constants; l, m, n are Murnaghan third order elastic constants

414

and δαβ is the Kronecker delta.

26

415

416 417 418 419 420 421

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422

[4] Z. Abiza, M. Destrade, R. W. Ogden, Large acoustoelastic effect, Wave Motion 49 (2) (2012) 364–374.

423

[5] D. S. Hughes, J. L. Kelly, Second-order elastic deformation of solids, Phys. Rev. 92 (5) (1953) 1145.

424

[6] R. A. Toupin, B. Bernstein, Sound waves in deformed perfectly elastic materials. acoustoelastic effect, J. Acoust. Soc.

425

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Peng Zuo: Conceptualization, Methodology, Software, Writing-Original draft preparation. Xudong Yu: Validation, Investigation. Zheng Fan: Supervision, Writing-reviewing and editing.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: