Nonlinear longitudinal waves in inhomogeneously predeformed elastic media

Abstract

The quasi one-dimensional problem of nonlinear longitudinal wave propagation in the elastic medium undergoing inhomogeneous plane prestrain is investigated theoretically. The analytical solution to describe the propagation of the wave with an arbitrary smooth initial profile is derived. The influence of the magnitude of the prestrain intensity on the distortion of the wave profile is studied. The sine-wave propagation in the medium subjected to the distributed static load is considered in more detail. The dependence of the sine-wave characteristics on the physical and geometrical properties of the medium and on the parameters of the predeformed state is clarified. The possibility to enhance the efficiency of ultrasonic nondestructive testing making use of the nonlinear effects of wave propagation is discussed. The algorithm to evaluation of the parameters of plane strain on the basis of wave profile evolution data is proposed.

Nondestructive characterization of materials with variable properties

“…(15) correct the solution taking the effects caused by the inhomogeneity (inhomogeneous predeformation) and nonlinearity into account. The amound of information available in NDT is dependent on the effecienty of this correction [44]. If the displacements of the material points caused by the wave motion are much smaller than the corresponding displacemets caused by predeformation, i.e.…”

“…The predeformed state (plain strain) is determined by the coeffiecients of polynomials that describe this state [19], [44]. These numerous coefficients are functions of the geometrical and physical properties of the material and the external load.…”

“…(13)) is excited in the material at X1 = 0, and it is recorded at the opposite surface X1 = h. Informative wave characteristics are amplitudes AI, A2 and phase shifts 0i, 02 of two first sine-wave harmonics. The harmonics phase velocities ~;~, cp ~ and the group velocity eg~ are derivable wave characteristics [44]. It is possible to evaluate the parameters of the predeformed state (coefficients a and b), external forces and physical properties of the material (density Q0, combination of linear elastic coefficients a, combination of nonlinear elastic coefficients ,3 (see Eq.…”

“…The phase and group velocities of the harmonics cp ~ and % may be calculated by the formulae [44] cp = ...... (22) C,ffc ~ (Od,~+l --(-g)/(g',~.+t --~:~) , (23) where 9,~ is the wavenumber of the n-th harmonic.…”

Non-linear interaction of waves in prestressed material

“…This case was studied in [20]. The conclusion was that the non-linear effects of wave motion contain maximum information about the prestressed state provided the displacements caused by prestress and wave motion are of the same order.…”