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This paper presents the effects of surface effects in the cavity of variable curvature. The wave function expansion method and the conformal mapping method are used in the solution of dynamic stress concentration factor around an irregularly shaped cavity at nano-scale. The stress boundary conditions on the surface are obtained by using the generalized Young-Laplace equation. The results show that the degree of stress concentration becomes more obvious with curvature increasing. Taking the elliptical cavity as an example, the influence of the ration of the major and minor axis of the ellipse, the numbers of the incident wave and the surface effects on the dynamic stress concentration factor are analyzed. The ration of the major and minor axis, the incident wave frequency and the surface effects show the pronounced effects on the dynamic stress concentration distributions.

Nano-structured materials and nano-sized structural elements have unique mechanical and physical properties. Atoms near the surface and interface of solids have different equilibrium positions and energies from that of the bulk material. Therefore, the surface and interface of solids display different mechanical properties compared with their bulk parts. To account for the effects of surface or interface, Gurtin et al. established the surface elasticity theory, which a surface is regarded as a negligibly thin membrane adhered to the bulk without slipping. The presence of surface stress gives rise to a non-classical boundary condition, which shows a good agreement with directly atomic simulations [

In fact, the surface elasticity theory can be applied not only to the statics analysis, but also to the dynamics analysis. In the framework of surface elasticity theory, the scattering of plane compressional and shear waves by a single nano-sized coated fiber and the multiple scattering by two cylinder inclusions, which embedded in an elastic matrix is studied by Ou and Lee using the method of eigenfunction expansion [

The above only consider the effects of surface effects in the case of constant curvature. However, in the preparation process of materials, cracks, inclusions, holes and other defects with variable curvature are often brought, such as elliptical and ellipsoidal cavities. Liu et al. [

Assume a harmonically time-varying plane P wave propagating at an angle α with respect to the x-axis with circular frequencies ω as shown in

Such an incident wave can be described by a displacement potential in the x-yplane as

ϕ inc ( x , y ) = ϕ 0 exp [ I k p ( x cos ( α ) + y sin ( α ) ) ] . (1)

where ϕ 0 is the amplitude of incident wave, the superscript inc represents the incident wave, k p = ω / c p is the wave number of P wave, c p = ( λ + 2 μ ) / ρ is the wave velocity of P wave, I = − 1 . For the convenience of calculation, the time-dependent term e − I ω t is suppressed throughout.

Based on the complex function theory, we introduce complex variables z = x + I y , Equation (1) is then transformed to the following form

ϕ inc ( z , z ¯ ) = ϕ 0 exp [ I k p 2 ( z e − I α + z ¯ e I α ) ] . (2)

When the incident wave impinges upon the irregularly shaped cavity boundary, two waves are reflected from the boundary [

ϕ ref ( z , z ¯ ) = ∑ n = − ∞ ∞ A n H n ( 1 ) ( k p | z | ) ( z | z | ) n , (3)

ψ ref ( z , z ¯ ) = ∑ n = − ∞ ∞ B n H n ( 1 ) ( k s | z | ) ( z | z | ) n . (4)

where the superscript ref represents the reflect wave, A n and B n are arbitrary expansion coefficients to be determined. k s = ω / c s is the wave number of SV wave, c s = μ / ρ is the wave velocity of SV wave. H n ( 1 ) ( ⋅ ) is the nth Hankel function of the first kind. Hankel function is related to the Bessel functions of the first kind J n ( ⋅ ) and the second kind Y n ( ⋅ ) by H n ( 1 ) ( ⋅ ) = J n ( ⋅ ) + I ⋅ Y n ( ⋅ ) .

In this section, we use the variable η = ζ + I ξ . For wave scattering problems involving noncircular boundary [

The mapping function ω ( η ) of the following form [

z = ω ( η ) = c 0 η + ∑ n = 1 ∞ c n 1 η n . (5)

Substituting Equation (5) into Equations (2)-(4), we find

ϕ inc ( η , η ¯ ) = ϕ 0 exp { I k p 2 [ ω ( η ) e − I α + ω ( η ) ¯ e I α ] } , (6)

ϕ ref ( η , η ¯ ) = ∑ n = − ∞ ∞ A n H n ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n , (7)

ψ ref ( η , η ¯ ) = ∑ n = − ∞ ∞ B n H n ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n . (8)

Thus the total elastic field is described by ϕ = ϕ inc + ϕ ref and ψ = ψ ref .

Meanwhile, the stress components σ r r , σ θ θ and σ r θ in η-plane are given by [^{ }

σ r r = − k p 2 ( λ + μ ) ϕ + 2 μ η 2 r 2 ω ′ ( η ) ¯ ∂ ∂ η [ 1 ω ′ ( η ) ∂ ∂ η ( ϕ + I ψ ) ] + 2 μ η ¯ 2 r 2 ω ′ ( η ) ∂ ∂ η ¯ [ 1 ω ′ ( η ) ¯ ∂ ∂ η ¯ ( ϕ − I ψ ) ] , (9)

σ θ θ = − k p 2 ( λ + μ ) ϕ − 2 μ η 2 r 2 ω ′ ( η ) ¯ ∂ ∂ η [ 1 ω ′ ( η ) ∂ ∂ η ( ϕ + I ψ ) ] − 2 μ η ¯ 2 r 2 ω ′ ( η ) ∂ ∂ η ¯ [ 1 ω ′ ( η ) ¯ ∂ ∂ η ¯ ( ϕ − I ψ ) ] , (10)

σ r θ = 2 I μ η 2 r 2 ω ′ ( η ) ¯ ∂ ∂ η [ 1 ω ′ ( η ) ∂ ∂ η ( ϕ + I ψ ) ] − 2 I μ η ¯ 2 r 2 ω ′ ( η ) ∂ ∂ η ¯ [ 1 ω ′ ( η ) ¯ ∂ ∂ η ¯ ( ϕ − I ψ ) ] . (11)

Assume that the surface adheres perfectly to the bulk material without slipping. The equilibrium equations and the constitutive relations on the surface are expressed as

t α + σ β α , β s = 0 , (12)

σ i j n i n j = σ α β s κ α β , (13)

σ α β s = τ 0 δ α β + 2 ( μ s − τ 0 ) δ α γ ε γ β + ( λ s + τ 0 ) ε γ γ δ α β . (14)

where n i denotes the normal vector of the surface, t α is the negative of the tangential component of the traction t i = σ i j n j in the x α direction, and κ α β is the curvature of the surface. Two dimensional indicators α , β value 1, 2. The elastic property of anisotropic surface is characterized by two material constants μ s and λ s . σ α β s is surface stress tensor. τ 0 is the residual surface tension under unstrained condition. Generally, the surface inertia force can be neglected in Equation (14) for the dynamic problems.

In the bulk, the classical elastic theory holds, and the isotropic constitutive relation reads as

σ i j = 2 μ ( ε i j + ν 1 − 2 ν ε k k δ i j ) , (15)

For plane strain problems, the strain ε θ θ is given by

ε θ θ = 1 2 μ [ ( 1 − ν ) σ θ θ − ν σ r r ] , (16)

Thus the surface stress σ θ θ s is obtained as

σ θ θ s = τ 0 + E s ε θ θ . (17)

with E s = 2 μ s + λ s − τ 0 . Since the residual surface stress τ 0 always induces an additional deformation field, independent of the external loading, we assume the residual surface stress τ 0 to be zero in what follows.

The boundary conditions on the circular hole ( r = a ) with surface effects are given from Equations (12) and (13) as

σ r r = σ θ θ s a , σ r θ = − ∂ σ θ θ s a ∂ θ . (18)

Thus in the η-plane, the boundary conditions on the unit circle with surface effects are given by

σ r r = σ θ θ s = E s 2 μ [ ( 1 − ν ) σ θ θ − ν σ r r ] = E s 2 μ { ( 1 − ν ) [ − 2 k p ( λ + μ ) ϕ − σ r r ] − ν σ r r } = − E s 2 μ [ 2 ( 1 − ν ) k p 2 ( λ + μ ) ϕ + σ r r ] , (19)

σ r θ = − ∂ σ θ θ s ∂ θ = E s 2 μ ∂ ∂ θ [ 2 ( 1 − ν ) k p 2 ( λ + μ ) ϕ + σ r r ] . (20)

Substituting of Equations (6)-(8) and Equations (9)-(11) into Equations (19) and (20), we find

∑ j = 1 2 ( ∑ n = − ∞ ∞ E i j n X j n ) = Y i , i = 1 , 2. (21)

where X 1 n = A n , X 2 n = B n , and the elements of E i j n are presented in Appendix A.

Multiplying both side of Equation (21) by e − I l θ and integrating between the interval ( − π , π ) , we have

∑ j = 1 2 ( ∑ n = − ∞ ∞ E i j l n X j n ) = Y i l , i = 1 , 2 ; l = 0 , ± 1 , ⋯ (22)

where E i j l n = 1 2 π ∫ − π π E i j n e − I l θ d θ , Y i l = 1 2 π ∫ − π π Y i e − I l θ d θ .

We can get a set of infinite algebraic equations to determine the coefficients A n and B n . After solving the linear equation system, the coefficients A n and B n can be obtained, then the elastic scattering fields due to the incident P wave can be determined.

The dynamic stress concentration factor (DSCF) is defined as the ratio between the hoop stress along the circumference of a cavity and the maximum amplitude of the incident stress wave at the same point

DSCF = | σ θ θ σ 0 | . (23)

For compressive incident wave, the maximum amplitude of incident stress σ 0 is

σ 0 = − μ k s 2 ϕ 0 . (24)

To illustrate the computational procedure, we consider the well-known case of a planar steady state compressive wave incident on a circular cavity. For a circular boundary, we take the mapping function as

z = ω ( η ) = a η . (25)

Assume a harmonic plane P wave propagating in the x-axis positive direction. By truncating the infinite matrix in Equation (22) to n = 3 and l = 3 . Let s = E s / 2 μ a . The Poisson ration is assumed to be ν = 0.25 for all calculation.

For low frequency incident wave with k p a = 0.2 , surface parameter s has significant effect on DSCF near a circle cavity. With the increase in parameter s, the DSCF continuously decrease around θ = π / 2 but increases around θ = 0 and θ = π in

For simplicity, we assume that the incident planar compressive wave is coming along the direction of the x-axis. The distribution of dynamic stress concentration in the neighborhood of an elliptic cavity, with semi-major axis a and semi-minor axis b can be made by choosing another mapping function

z = ω ( η ) = a + b 2 η + a − b 2 η ¯ . (26)

For a high frequency incident wave with k p a = π , the dynamic stress concentration factor at the elliptical cavity surface is more complicated and the images appear multiple asymmetric peaks in

In this paper, we have presented a method of complex functions to analyze the two dimensional problems of dynamic elasticity with non-circular geometry. The method is applied to evaluate the dynamic stress concentration factor around a circular and an elliptical cavity. Various type of scattering phenomena and dynamic stress concentrations due to a more general type of elastic inclusion can also be analyzed similarly.

The supports from the National Natural Science Foundation (Grant No. 11362009 and No. 11862014) are acknowledged.

The authors declare no conflicts of interest regarding the publication of this paper.

Ou, Z.Y. and Han, H.J. (2019) Surface Effects on the Diffraction of P Wave by an Arbitrary Shaped Cavity. Open Journal of Applied Sciences, 9, 439-450. https://doi.org/10.4236/ojapps.2019.96036

E 11 n = ( 1 + E s 2 μ ) { − k p 2 ( λ + μ ) H n ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + μ k p 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 + μ k p 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2 } + E s μ ( 1 − ν ) k p 2 ( λ + μ ) H n ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n (A.1)

E 12 n = ( 1 + E s 2 μ ) { I μ k s 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 − I μ k s 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2 } (A.2)

Y 1 = { ( 1 + E s 2 μ ) [ λ + μ + μ η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ e − 2 I α + μ η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) e 2 I α ] − E s μ ( λ + μ ) ( 1 − ν ) } k p 2 ϕ 0 exp { I k p 2 [ ω ( η ) e − I α + ω ( η ) ¯ e I α ] } (A.3)

E 21 n = I μ k p 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 − I μ k p 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2 + E s 2 μ ∂ ∂ θ F 1 (A.4)

E 22 n = − μ k s 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 − μ k s 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2 + E s 2 μ ∂ ∂ θ F 2 (A.5)

Y 2 = [ I μ η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ e − 2 I α − I μ η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) e 2 I α ] k p 2 ϕ 0 × exp { I k p 2 [ ω ( η ) e − I α + ω ( η ) ¯ e I α } − E s 2 μ ∂ ∂ θ F 3 (A.6)

F 1 = ( 2 ν − 1 ) k p 2 ( λ + μ ) H n ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − μ k p 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 (A.7)

− μ k p 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k p | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2

F 2 = − I μ k s 2 η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ H n − 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n − 2 + I μ k s 2 η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) H n + 2 ( 1 ) ( k s | ω ( η ) | ) [ ω ( η ) | ω ( η ) | ] n + 2 (A.8)

F 3 = [ ( 2 ν − 1 ) ( λ + μ ) + μ η 2 ω ′ ( η ) 2 ω ′ ( η ) ¯ e − 2 I α + μ η ¯ 2 ω ′ ( η ) ¯ 2 ω ′ ( η ) e 2 I α ] k p 2 ϕ 0 × exp { I k p 2 [ ω ( η ) e − I α + ω ( η ) ¯ e I α ] } (A.9)