Guided Waves in a Solid Rod Embedded in Infinite Medium with Weak Interface

The dispersion relation of the infinite long rod model with a weak interface has been deduced via wave equation of the guided waves and boundary spring interface model. The dispersion characteristics of the rigid interface and the slip interface are calculated, while the effect of different axial stiffness coefficient is also discussed. The results suggest that the phase velocity of the slip interface decreases slowly with increasing frequency at the low frequency of the L (0,1) mode. As for the weak interface, when the axial coefficient Kt is in the particular range of [10,10]pa/m, the interface turns from the slip interface to the rigid interface. That shows the value of axial coefficient Kt is an essential prerequisite for the non-destructive evaluation to evaluate the interface cementation degree.


Introduction
Propagation of guided waves in a solid rod embedded in infinite medium with weak interface is an interesting research topic of practical importance. For example, it is urgent to apply Non-destructive evaluation (NDE) for the rock bolts in the mixture of rock and concrete, which are inserted to ensure the stability of the groundwork. One key issue in the above mentioned NDE process is how to evaluate bonding state between rock bolt and concrete by guided ultrasonic waves [1][2][3][4] .
The propagation of guided wave in solid cylindrical media has already been widely studied, but most work is based on a model of a multilayered columnar structure with continuous interface [5][6][7][8] , few attention has been focused on the semi-infinite space structure with discontinuous interface [9][10] .
In this paper, we study the propagation characteristics of guided waves in a solid rod embedded in infinite medium with weak interface. This paper is arranged as follows. First, the dispersion equation using spring interface model is deduced. Second, the dispersion characteristics of guided waves in the structure under the rigid interface and slip interface are discussed, respectively. Third, we investigated the effects of different axial coefficient on the guided waves under the weak interface.

Weak Interface
The interface between different solid media of the layered structure is inevitable, and can be divided into two  [7] is such an interface that can withstand the normal (or radial) stress perpendicular to the interface while the tangential (or axial) stress is identical to zero.
That is, the normal (or radial) stress and displacement are continuous; the tangential (or axial) displacement is discontinuity [11] .
In the procedure of using complex media or composite material, the micro-cracks occur at the interface, and the mechanical strength is reduced to form a weak boundary layer. The spring model can describe the physical characteristics of a weak interface, thus the boundary conditions of mechanical quantity at the interface is available [11] . layer. The potential function can be expressed as

Dispersion Equation
The potential function satisfies the wave equation, and can be described as where, Cl and Cs are the P-wave velocity and S-wave velocity, respectively. t is time, and 2 is Laplace Here, λ and μ are both Lame parameters, ρ is the density.
It is convenient to introduce displacement potentials.
The normal and tangential stress tractions are related to displacements using the Hooke's law.
The radial and axial displacement components ur and uz can be expressed by substituting Eqs. (1) and (2)

10)
where, Kn and Kt are the radial stiffness coefficient, and the axial stiffness coefficient, respectively. The functions of Kn and Kt are composed of the interface layer thickness h, equivalent bulk modulus K0 and shear modulus G0 [11] . They are expressed as , the boundary condition When the right-hand side of Eq.(13) is zero matrix, the necessary and sufficient condition for the non-zero solutions of the equation is the coefficient . It is the dispersion equation whose solution corresponds to the guided wave.

Numerical results
Since our work aims at the rock bolt detection, our simulation selects concrete as material of infinite medium, and steel as the solid rod. By setting the radius of solid rod to be 10mm so that the dispersion characteristics of the structure under rigid interface, slip interface and different axial stiffness coefficient are analysed.

Rigid Interface
As the stiffness coefficient , Kn Kt o f , the interface is the rigid interface, i.e., the stress and displacement are continuous.
The phase velocity dispersion curves are shown in the  As for the spring model, the axial elasticity coefficient is a measure of weak interface to withstand axial stress.

Weak Interface
Therefore, the dispersion curves under different axial stiffness coefficient are calculated to evaluate the weak interface.