Wave propagation in coated cylinders with reference to fretting fatigue

Wave propagation in coated cylinders with reference to fretting fatigue

M RAMESH^∗, SATISH V KAILAS and K R Y SIMHA Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012

e-mail: mramesh@mecheng.iisc.ernet.in

Abstract. Fretting fatigue is the phenomenon of crack initiation due to dynamic contact loading, a situation which is commonly encountered in mechanical cou- plings subjected to vibration. The study of fretting fatigue in high frequency regime has gained importance in recent years. However the stress wave effects at high fre- quency loading is scanty in the literature. The objective of present investigation is to study stress wave propagation in cylinders with reference to high frequency fretting. The case of a coated cylinder is considered since coating is often provided to improve tribological properties of the component. Rule of mixtures is proposed to understand the dispersion phenomenon in coated or layered cylinder knowing the dispersion relation for the cases of homogeneous cylinders made of coating and substrate materials separately. The possibility of stress wave propagation at the interface with a particular phase velocity without dispersion is also discussed.

Results are given for two different thicknesses of coating.

Keywords. High frequency fretting; stress waves; frequency equation; layered cylinder.

1. Introduction

Fretting fatigue is the phenomenon of crack initiation due to dynamic contact loading. Such situations are commonly encountered whenever materials under contact are subjected to vibration. The frequency of vibration or contact loading is an important parameter. The general frequency regime for fretting fatigue is considered to be 20 Hz to 20 KHz. The literature on fretting fatigue in the high frequency regime is generally limited, except in the recent years. The motivation for studying of fretting fatigue at higher frequency is to investigate the implications of contact dynamics on fretting and fatigue lives and the evolution of coefficient of friction (Matlik & Farris 2003; Farris et al 2003). Also fretting fatigue tests at high frequencies provide an alternative method for conventional testing by accelerating the testing process, which otherwise would take considerably longer time. Soderberg et al (1986), studied fretting wear behaviour of austenitic stainless steel under ultrasonic frequency for the purpose of

∗For correspondence

339

accelerating the test and compared the results with that obtained for low frequencies. Rehbein et al (1999), developed high frequency tribometers to study tribological behaviour of friction materials under ultrasonic fretting conditions.

Understanding high frequency fretting fatigue in shafts, rods and piping requires sophis- ticated elastodynamic techniques of stress analysis. In general, both longitudinal and shear waves are excited in high frequency fretting. The relative importance of these two waves depends on fretting frequency besides geometrical factors and material properties. Thus, pre- dicting initiation and propagation of micro cracks during fretting demands a proper study of stress wave propagation in cylinders and tubes. Pochhammer made fundamental contribution in this area in 1876 (Graff 1975). This analysis is an extension of simple one-dimensional wave propagation in thin bars. Depending on the size of the cylinder and presence of anti fretting coatings if any, stress waves of different frequencies propagate at different phase velocities. The velocity of stress wave propagation determines the rate of energy trans- port imparted by the dynamic contact loading. Since crack initiation is one of the means of energy dissipation, stress wave characteristics control crack initiation sites and their extension.

1.1 Wave propagation in cylinders

Most of the industrial components subjected to fretting have axisymmetric geometry for example shrink-fitted shaft. Hence focus in this work is given to thin rods and cylinders. The equilibrium equation for thin rods whose displacementuat a given axial distancexat time instantt is given by

E∂²u

∂x² =ρ∂²u

∂t² ⇒ ∂²u

∂x² = 1 c₀²

∂²u

∂t², (1)

wherec₀=

E

ρ, Eis the Young’s modulus andρis the density of the material.

The valuec0is a material constant and is called the bar wave speed. Waves of all frequencies travel at this speed in a thin cylinder or rod. However when the cylinder radius is larger, waves of different frequencies travel with different velocities, the characteristics of which will be given by Pochhammer frequency equation. This phenomenon is called as dispersion. Disper- sion is not considered in the preliminary analysis for bar wave speed. The bar wave concept applies only to thin cylinders. For finite cylinders the Poisson effect demands Pochhammer analysis. Pochhammer analysis highlights the dispersive behaviour of stress wave propaga- tion in cylinders. Accordingly different frequencies of stress waves propagate at different phase speeds. A plot showing the phase velocity (c0) as the function of wavenumber(ξ )will provide the dispersive characteristics of wave propagation. The phase velocityc= ^ω_ξ where ωis the angular frequency andξ = wave length^2π is the wave number. The phase velocityccor- responds to the velocity at which the wave of wavenumberξ travels in the solid. Figure 1 shows the dispersive characteristics of a cylinder in terms of normalised phase velocity _c^c

0

as a function of normalised wavenumber ^{ξ R}_2π the result given by Pochammer in 1876. In the analysis wave is assumed to propagate only along the axial direction and not in the radial and angular directions. The plot shows that in a cylinder waves of small wavenumber/large wave- length travel at phase velocityc0and waves of large wavenumber/small wavelength travel at a speedc_Rcalled the Rayleigh speed. Further, waves of small wavenumber/large wavelengths correspond to waves of low frequency and waves of large wavenumber/small wavelengths

Figure 1. Dispersion relation for homogeneous cylinder.

correspond to waves of high frequency. Hence an equivalent statement can be made as waves of low frequencies propagate at velocity c₀ while waves of high frequency propagate at velocityc_R.

Rayleigh wave corresponds to the surface wave propagation in a semi infinite half space subjected to traction free condition at the surface. The speed of Rayleigh wave is the low- est compared to dilatational, shear and bar wave speeds and hence carry more energy. The Rayleigh wave speedc_Rin a material can be obtained from the relation (Graff 1975)

c_R c2

6

−8 c_R

c2

4

+(24−16k⁻²) c_R

c2

2

−16(1−k⁻²)=0, (2)

where c2 =

μ

ρ is the shear wave speed and k² = ^2(1−ν)_1−2ν with μ, ρ and ν being the shear modulus, density and poisson’s ratio of the material, respectively. Ifν = ¹₄ then ^c_c^R

2 =

2− ^√2₃¹₂

=0·9194.

1.2 Frequency equation for composite cylinders

Whenever a material has good mechanical properties but poor fretting resistance it is advan- tageous to provide a layer of material having good fretting resistance instead of changing the entire component material. This extra layer of material can be provided by coating, spraying or by any other surface treatment methods. Okane et al (2003) showed that providing WC–

Co layer over NiCrMo steel retards fretting crack initiation and reduces crack propagation rate. Similar results were obtained by Nishida et al (2003) for the case of aluminium alloy provided with anodic film. The inhomogeneity introduced by such coating can have signifi- cant influence on the wave propagation if the coating is considerably thicker and the material properties differ significantly. Also, presence of any interface in general alters the nature of stress wave propagation.

Whittier & Jones (1967) studied axisymmetric wave propagation in a two-layered cylinder and compared the results obtained with the shell theory. They also mentioned about possibility of Stonely wave at the interface. They concluded that for the case of large wave numbers the phase velocity is decreasing, and approaches from above the Rayleigh wave velocity of

Figure 2. Composite cylinder with coordinate system.

the slower medium. Wave propagation in a coated cylinder is addressed in this investigation (Ramesh et al 2006).

For analysis a cylinder with coating is considered as a composite cylinder with the outer shell made of different material. LetR1andR2are the outer and inner radii of the composite cylinder respectively. The composite cylinder with the corresponding cylindrical coordinate system is shown in figure 2.

The formulation is similar to that used by Whittier & Jones (1967) except that inner cylinder is considered to be solid instead of hollow. The mathematical formulation for dynamic stress analysis is obtained using dynamic potentialsand which are governed by the following equations

∇²= 1 c²₁

∂²

∂t² ; ∇² = 1 c²₂

∂²

∂t² , (3)

where

∇²= ∂²

∂r² +1 r

∂

∂r + ∂²

∂z². (4)

Letis the dilatation and is given by = ∂ur

∂r +ur

r +∂uz

∂z , (5)

whereur anduzare the radial and axial displacements.

The following expressions give the stresses and displacements in terms of potential func- tions.

2μu_r = ∂

∂r + ∂²

∂r∂z (6)

2μu_z= ∂

∂z +∂²

∂z² − 1 c²₂

∂²

∂t² (7)

2μ= ∇² (8)

σ_r =λ+2μ∂u_r

∂r (9)

σz=λ+2μ∂uz

∂z (10)

σ_θ =λ+2μur

r (11)

τrz=μ ∂ur

∂z +∂uz

∂r

(12) Assuming

(r, z, t )=(r, z)sin(ωt ) (13)

(r, z, t )= (r, z)sin(ωt ). (14)

the time factor sin(wt )is suppressed in the sequel.

The governing equations 3 become

∇²= −ω²

c₁²; ∇² = −ω²

c²₂ . (15)

The method of Fourier integral transform is used to solve forand . Considering the case of symmetry ofσrand anti symmetry ofτrzaboutz=0 the dynamic potentialsand are chosen to be even and odd functions respectively. Hence Fourier cosine transform is applied overand Fourier sine transform is applied over as given below.

(r, ξ )= ^∞

0

(r, z)cos(ξ z)dz (16)

(r, ξ )= ^∞

0

(r, z)sin(ξ z)dz. (17)

The inverse transforms are:

(r, z)= 2 π

∞ 0

(r, ξ )cos(ξ z)dξ (18)

(r, z)= 2 π

∞ 0

(r, ξ )sin(ξ z)dξ. (19)

The stresses and displacements in the Fourier space are:

2μur = d

dr +ξd

dr (20)

2μu_z= −ξ + ω²

c²₂ −ξ²

(21)

2μ= −ω²

c²₁ (22)

σr = − λ 2μ

ω²

c²₁+d² dr² +ξd²

dr² (23)

τ_rz= −ξd dr + 1

2 ω²

c²₂ −2ξ² d

dr (24)

σ_θ = − λ 2μ

ω² c²₁+1

r d

dr +ξd dr

(25)

σ_z= − λ 2μ

ω² c²₁+ξ

−ξ + ω²

c₂² −ξ²

. (26)

Applying the respective transform over the governing equations of the potentials yields the following ordinary differential equations.

d² dr² + 1

r d

dr + ω²

c²₁ −ξ²

=0 (27)

d² dr² +1

r d

dr + ω²

c₂² −ξ²

=0. (28)

The solution to the above equations corresponding to outer layer of cylinder made of materialAis

Case (i) if 0< ξ² < ^ω2

c²_1A thenα_A² = _c^ω2²

1A −ξ²,β_A² = _c^ω2² 2A−ξ²

A=A1(ξ )J0(αAr)+A2(ξ )Y0(αAr) (29)

A= A₃(ξ )

β_A J₀(β_Ar)+ A₄(ξ )

β_A Y₀(β_Ar). (30)

Case (ii) if ^ω2

c²_1A < ξ² < _c^ω2² 2A

thenα_A² =ξ²−_c^ω22

1A

,β_A² = _c^ω22

2A−ξ²

_A=A₁(ξ )I₀(α_Ar)+A₂(ξ )K₀(α_Ar) (31)

A= A3(ξ ) βA

J0(βAr)+ A4(ξ ) βA

Y0(βAr). (32)

Case (iii) if ^ω2

c²_1A < ξ² <∞thenα_A² =ξ²−_c^ω2² 1A

,β_A² =ξ²− _c^ω2² 2A

A=A1(ξ )I0(αAr)+A2(ξ )K0(αAr) (33)

A= A₃(ξ )

β_A I0(β_Ar)+A₄(ξ )

β_A K0(β_Ar), (34)

wherec1A=

λA+2μA

ρA andc2A=

μ

ρ are the dilatational and shear wave speeds respectively in materialA. The functionsJ0, Y0, I0andK0are the Bessel functions. The solution for inner core made of materialBis obtained by replacing suffixAbyB. The constantsB2 =B4=0 the core being solid. For the case of homogeneous cylinder made of materialAwhich corresponds to the Pochhammer problem A2 = A4 = 0. However, a general case of coated cylinder requires six non-zero constants.

Using this solution forA, B, Aand Bthe stresses and displacements in the Fourier space are:

CylinderA(coating):

2μ_Au_rA =A1(ξ )a11(r)+A2(ξ )a12(r)+A3(ξ )a13(r)+A4(ξ )a14(r) 2μ_Au_zA=A₁(ξ )a₂₁(r)+A₂(ξ )a₂₂(r)+A₃(ξ )a₂₃(r)+A₄(ξ )a₂₄(r) σrA =A1(ξ )a31(r)+A2(ξ )a32(r)+A3(ξ )a33(r)+A4(ξ )a34(r) τ_rzA=A₁(ξ )a₄₁(r)+A₂(ξ )a₄₂(r)+A₃(ξ )a₄₃(r)+A₄(ξ )a₄₄(r) σθA =A1(ξ )a51(r)+A2(ξ )a52(r)+A3(ξ )a53(r)+A4(ξ )a54(r)

σzA=A1(ξ )a61(r)+A2(ξ )a62(r)+A3(ξ )a63(r)+A4(ξ )a64(r). (35) CylinderB(core):

2μ_BurB=B1(ξ )b11(r)+B3(ξ )b13(r) 2μ_Bu_zB =B1(ξ )b21(r)+B3(ξ )b23(r) σrB=B1(ξ )b31(r)+B3(ξ )b33(r) τrzB =B1(ξ )b41(r)+B3(ξ )b43(r) σ_{θ B} =B₁(ξ )b₅₁(r)+B₃(ξ )b₅₃(r)

σzB =B1(ξ )b61(r)+B3(ξ )b63(r). (36) The expressions fora_ij and b_ij are given in Appendix A. The boundary conditions for the frequency equation of homogeneous cylinder is thatσr =τrz =0 atr =R, where R is the radius of cylinder. For the case of composite cylinder the stress and displacement continuity has to be ensured at the interface along with the traction free surface condition.

Substituting the expressions for the transformed stresses and displacements from equa- tions 35 and 36 in the above mentioned boundary conditions we get the following equation in the matrix form

[M]6×6[A]6×1=[0]6×1, (37)

where

M=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

a₃₁(R₁) a₃₂(R₁) a₃₃(R₁) a₃₄(R₁) 0 0 a41(R1) a42(R1) a43(R1) a44(R1) 0 0 a31(R2) a32(R2) a33(R2) a34(R2) −b31(R2) −b33(R2) a₄₁(R₂) a₄₂(R₂) a₄₃(R₂) a₄₄(R₂) −b₄₁(R₂) −b₄₃(R₂) a11(R2) a12(R2) a13(R2) a14(R2) −^μ_μ^A_Bb11(R2) −^μ_μ^A_Bb13(R2) a21(R2) a22(R2) a23(R2) a24(R2) −^μ_μ^A_Bb21(R2) −^μ_μ^A_Bb23(R2)

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

A=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣ A1

A2

A₃ A4

B1

B3

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

; 0=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣ 0 0 0 0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦ ,

where A is the column vector with the constants Aj and Bj as the elements, matrix M consists of the coefficients of these constants as the elements and 0 represents the zero column vector. The frequency equation is now given by Det(M)=0. For the homogeneous case the frequency equation (Pochhammer frequency equation (Graff 1975)) becomes

2α

R (β²+ξ²)J1(αR)J1(βR)−(β²−ξ²)²J0(αR)J1(βR)

−4ξ²αβJ₁(αR)J₀(βR)=0. (38)

The expression for composite cylinder case being more complicated is not shown here.

However frequency equation for the case of composite cylinder is solved numerically and the results are compared with that of the homogeneous case.

2. Results and discussion

The dispersion characteristics for the case of homogeneous cylinder (figure 1) shows that at lower wave number and hence at lower frequencies the propagation speed corresponds to that of bar wave speed. However at higher wave number and hence at higher frequencies the speed of wave propagation is same as that of Rayleigh wave speed. This shows that at higher frequencies the wave propagation and the energy transfer occur predominately at the surface.

Since fretting being a near surface phenomenon these surface waves will have considerable effect on the fretting fatigue. For the case of composite cylinder or coated cylinder different combinations of material properties are chosen. IfλA = μA andλB = μB then Poisson’s ratioν_A=ν_B =1/4. Also density of both the materials is taken to be equal to 2500 Kg/m³. The value ofμ_B is taken to be 25×10⁹. Results are shown for the cases ofλ_A =λ_B, 2λ_B

Figure 3. Effect of elastic mis- match between outer layer and inner core material.

and 4λ_B, the interface radiusR₂=0·5 (figure 3). The phase velocity is normalised with the bar wave speed of the coating. The normalised phase velocityc/c_0Ais plotted with respect to the normalised wave numberξ = ^{ξ R}_2π¹. The nature of plot for the case of coated cylinder is similar to that of homogeneous cylinder except for two reasons. First one is that the plot is shifted down by almost a constant value depending upon the ratioλ_A/λ_B. Also they have small maxima and minima whereas the plot for homogeneous case monotonically decreases to Rayleigh wave speed of the material.

This similarity in the nature of the curves infers that the approximate plot for the case of composite cylinder can be obtained from the plots corresponding to substrate and coating material by using the rule of mixture (figure 4). The rule of mixture can be appropriately stated as

ccomposite=xcsubstrate+(1−x)ccoating, (39)

wherex= ^R_R²²2 1

ccomposite=Phase velocity in the coated cylinder of outer radiusR1and interface radiusR2

ccoating=Phase velocity in a homogeneous cylinder of radiusR1completely made of coating materialA

csubstrate = Phase velocity in a homogeneous cylinder of radius R1 completely made of substrate (core cylinder) materialB.

Figure 4 shows the plot of _c^c

0A vsξfor the cases of composite, substrate and coating. Also the plot for the composite calculated from the rule of mixtures is shown in dotted lines.

The plot shows that the prediction by rule of mixtures is good for the values ofξ R1/(2π ) near zero and near 1·5. In the intermediate region prediction by rule of mixtures is still fair.

However the rule of mixture fails to predict the small oscillation near^{ξ R}_2π¹ >1 in the dispersion curve of composite cylinder. Figure 5 shows the slope of dispersion curves plotted in figure 4.

The plot shows that the dispersion curve of composite cylinder crosses zero at two points near

ξ R1

2π =0·9 and^{ξ R}_2π¹ =1·4 which is not seen in the other two cases. This is important because

Figure 4. Prediction by rule of mixtures.

the group velocity depends on the slope of dispersion curve. Also this implies that there is a possible case of wave travelling with a particular phase velocity without dispersion.

The group velocity(cg)corresponds to the velocity at which the energy of the wave is being transmitted. The expression forc_g shows thatc_g = ^dω_dξ = c+ξ_dξ^dc becauseω = cξ.

Thuscgand hence the rate of energy transmission depends on the slope of dispersion curve

dc

dξ. In figure 6 group velocity of composite is plotted along with that of homogeneous cases made of the coating and core materials. The plot shows that behaviour ofc_gis very different for composite cylinder compared to the homogeneous cases. Hence the rule of mixture will not be able to predict thec_g of composite cylinders even though it predicts the dispersion curve fairly well. Further, it should be noted that thec_gcurve for composite cylinder oscillates whose amplitude is contained between the two curves correspond to the homogeneous cases made of materialsAandB.

Figure 5. Gradient of phase speed curve.

Figure 6. Group speed Vs Wave number.

The dispersion curve for coated cylinder is shown in figure 7 for the case of interface radius being 0·8R1and 0·9R1. The plot shows that the dispersion relation becomes similar to that of the core as the thickness of coating is reduced. This can be well explained by the rule of mixture.

3. Conclusion

The elastodynamics of high frequency fretting in coated or layered cylinder is investigated.

The effect of mismatch between elastic properties of the coating and substrate or core on dispersion is presented. The rule of mixtures is proposed to understand the dispersion rela- tion in layered cylinder, whose predictions are good both at lower and higher values of wave number except at the intermediate range, when compared with the actual dispersion relation for layered cylinder. The possibility of phase velocity without dispersion at the interface is

Figure 7. Effect of coating thick- ness.

demonstrated however the corresponding range of wave number is small. Results for disper- sion relation are shown for the cases of interface radius lying closer to the outer surface. The dispersion relation approaches the dispersion relation corresponding to that of the inner core as the interface radius becomes closer and closer to the surface. Since stress wave propagation and crack initiation are phenomena of energy transfer and dissipation a greater interaction is expected between them. Thus, at higher frequencies the contact dynamics and stress wave propagation will play a major role in fretting. Especially for the case where a thick coating is present the dynamic analysis becomes inevitable.

Appendix A

Appendix A.1. Expressions fora_ij Case (i)

0< ξ < _c^ω

1A < _c^ω

2A <∞

a₁₁= −α_AJ₁(α_Ar) (40)

a₁₂= −α_AY₁(α_Ar) (41)

a13= −ξ J1(βAr) (42)

a14= −ξ Y1(βAr) (43)

a21= −ξ J0(αAr) (44)

a22= −ξ Y0(α_Ar) (45)

a₂₃=β_AJ₀(β_Ar) (46)

a24=βAY0(βAr) (47)

a31= − λ_A

2μ_A +1

α_A² + λ_A 2μ_Aξ²

J0(α_Ar)+α_A²J₁(α_Ar)

α_Ar (48)

a₃₂= − λ_A

2μ_A +1

α_A² + λ_A 2μ_Aξ²

Y₀(α_Ar)+α_A²Y₁(α_Ar)

α_Ar (49)

a₃₃=ξβ_A

−J₀(β_Ar)+J₁(β_Ar) β_Ar

(50)

a₃₄=ξβ_A

−Y₀(β_Ar)+ Y₁(β_Ar) β_Ar

(51) (52)

a41=ξ αAJ1(αAr) (53)

a42=ξ α_AY1(α_Ar) (54) a43= −1

2(β_A² −ξ²)J1(βAr) (55)

a44= −1

2(β_A² −ξ²)Y1(βAr) (56)

a51= − λ_A

2μ_A(α_A² +ξ²)J0(α_Ar)−α_A²J₁(α_Ar)

α_Ar (57)

a₅₂= − λ_A

2μ_A(α_A² +ξ²)Y₀(α_Ar)−α²_AY₁(α_Ar)

α_Ar (58)

a53= −ξβA

J1(βAr)

βAr (59)

a54= −ξβ_AY₁(β_Ar)

β_Ar (60)

a₆₁= − λA

2μ_Aα²_A+ λA

2μ_A +1

ξ²

J₀(α_Ar) (61)

a₆₂= − λA

2μ_Aα²_A+ λA

2μ_A +1

ξ²

Y₀(α_Ar) (62)

a63=ξβAJ0(βAr) (63)

a64=ξβAY0(βAr). (64)

Case (ii) 0< _c^ω

1A < ξ < _c^ω

2A <∞

a11=αAI1(αAr) (65)

a12= −α_AK1(α_Ar) (66)

a₁₃= −ξ J₁(β_Ar) (67)

a₁₄= −ξ Y₁(β_Ar) (68)

a21= −ξ I0(αAr) (69)

a22= −ξ K0(αAr) (70)

a23=βAJ0(βAr) (71)

a24=β_AY0(β_Ar) (72)

a31= λ_A

2μ_A +1

α_A² − λ_A 2μ_Aξ²

I0(α_Ar)−α²_AI1(α_Ar)

α_Ar (73)

a32= λ_A

2μ_A +1

α_A² − λ_A 2μ_Aξ²

K0(α_Ar)+α_A²K₁(α_Ar)

α_Ar (74)

a33=ξβ_A

−J0(β_Ar)+J₁(β_Ar) β_Ar

(75)

a34=ξβ_A

−Y0(β_Ar)+ Y₁(β_Ar) β_Ar

(76)

a41= −ξ α_AI1(α_Ar) (77)

a42=ξ α_AK1(α_Ar) (78)

a43= −1

2(β_A² −ξ²)J1(βAr) (79)

a44= −1

2(β_A² −ξ²)Y1(βAr) (80)

a51= λ_A

2μ_A(α_A² −ξ²)I0(α_Ar)+α_A²I1(α_Ar)

α_Ar (81)

a₅₂= λ_A

2μ_A(α_A² −ξ²)K₀(α_Ar)−α_A²K₁(α_Ar)

α_Ar (82)

a53= −ξβA

J1(βAr)

βAr (83)

a54= −ξβ_AY1(β_Ar)

β_Ar (84)

a₆₁= λ_A

2μ_Aα_A² − λ_A

2μ_A+1

ξ²

I₀(α_Ar) (85)

a₆₂= λ_A

2μ_Aα_A² − λ_A

2μ_A+1

ξ²

K₀(α_Ar) (86)

a₆₃=ξβ_AJ₀(β_Ar) (87)

a₆₄=ξβ_AY₀(β_Ar). (88)

Case (iii) 0< _c^ω

1A < _c^ω

2A < ξ <∞

a11=αAI1(αAr) (89)

a12= −α_AK1(α_Ar) (90)

a₁₃=ξ I₁(β_Ar) (91)

a₁₄= −ξ K₁(β_Ar) (92)

a21= −ξ I0(α_Ar) (93)

a₂₂= −ξ K₀(α_Ar) (94)

a₂₃= −β_AI₀(β_Ar) (95)

a24= −βAK0(βAr) (96)

a31= λA

2μA +1

α_A² − λA

2μA

ξ²

I0(αAr)−α²_AI1(αAr)

αAr (97)

a32= λA

2μA +1

α_A² − λA

2μA

ξ²

K0(αAr)+α_A²K1(αAr)

αAr (98)

a33=ξβA

I0(βAr)−I1(β_Ar) βAr

(99)

a34=ξβA

K0(βAr)+K1(β_Ar) βAr

(100)

a₄₁= −ξ α_AI₁(α_Ar) (101)

a42=ξ αAK1(αAr) (102)

a₄₃= −1

2(β_A² +ξ²)I₁(β_Ar) (103)

a₄₄= 1

2(β_A² +ξ²)K₁(β_Ar) (104)

a₅₁= λ_A

2μ_A(α_A² −ξ²)I₀(α_Ar)+α_A²I₁(α_Ar)

α_Ar (105)

a52= λA

2μ_A(α_A² −ξ²)K0(αAr)−α_A²K1(αAr)

αAr (106)

a53=ξβA

I1(βAr)

βAr (107)

a₅₄= −ξβ_AK₁(β_Ar)

β_Ar (108)

a61= λ_A

2μ_Aα_A² − λ_A

2μ_A+1

ξ²

I0(α_Ar) (109)

a62= λ_A

2μ_Aα_A² − λ_A

2μ_A+1

ξ²

K0(α_Ar) (110)

a63= −ξβ_AI0(β_Ar) (111)

a₆₄= −ξβ_AK₀(β_Ar). (112)

Appendix A.2. Expressions forb_ij Case (i)

0< ξ < _c^ω

1B < _c^ω

2B <∞

b11= −αBJ1(αBr) (113)

b13= −ξ J1(βBr) (114)

b21= −ξ J0(αBr) (115)

b23=βBJ0(βBr) (116)

b31= − λB

2μB +1

α_B² + λB

2μB

ξ²

J0(αBr)+α²_BJ1(αBr)

αBr (117)

b33=ξβB

−J0(βBr)+ J1(βBr) βBr

(118)

b41=ξ α_BJ1(α_Br) (119)

b₄₃= −1

2(β_B² −ξ²)J₁(β_Br) (120)

b₅₁= − λ_B

2μ_B(α²_B+ξ²)J₀(α_Br)−α²_BJ₁(α_Br)

α_Br (121)

b₅₃= −ξβ_BJ₁(β_Br)

β_Br (122)

b₆₁= − λ_B

2μ_Bα_B² + λ_B

2μ_B +1

ξ²

J₀(α_Br) (123)

b63=ξβ_BJ0(β_Br). (124)

Case (ii) 0< _c^ω

1B < ξ < _c^ω

2B <∞

b11=α_BI1(α_Br) (125)

b₁₃= −ξ J₁(β_Br) (126)

b21= −ξ I0(αBr) (127)

b23=βBJ0(βBr) (128)

b31= λB

2μ_B +1

α²_B− λB

2μ_Bξ²

I0(αBr)−α_B²I1(αBr)

αBr (129)

b33=ξβB

−J0(βBr)+ J1(βBr) βBr

(130)

b41= −ξ αBI1(αBr) (131)

b₄₃= −1

2(β_B² −ξ²)J₁(β_Br) (132)

b51= λ_B 2μB

(α_B² −ξ²)I0(αBr)+α_B²I1(α_Br)

αBr (133)

b₅₃= −ξβ_BJ₁(β_Br)

β_Br (134)

b61= λB

2μB

α_B² − λB

2μB +1

ξ²

I0(αBr) (135)

b63=ξβBJ0(βBr). (136)

Case (iii) 0< _c^ω

1B < _c^ω

2B < ξ <∞

b₁₁=α_BI₁(α_Br) (137)

b₁₃=ξ I₁(β_Br) (138)

b21= −ξ I0(αBr) (139)

b23= −β_BI0(β_Br) (140)

b31= λ_B

2μ_B +1

α²_B− λ_B 2μ_Bξ²

I0(α_Br)−α_B²I1(α_Br)

α_Br (141)

b33=ξβ_B

I0(β_Br)−I₁(β_Br) β_Br

(142)

b41= −ξ αBI1(αBr) (143)

b₄₃= −1

2(β_B² +ξ²)I₁(β_Br) (144)

b₅₁= λ_B

2μ_B(α_B² −ξ²)I₀(α_Br)+α_B²I₁(α_Br)

α_Br (145)

b53=ξβB

I1(βBr)

βBr (146)

b61= λB

2μ_Bα_B² − λB

2μ_B +1

ξ²

I0(αBr) (147)

b63= −ξβBI0(βBr). (148)

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