Propagation of longitudinal and transverse waves in a multimodulus elastic medium

Abstract

UDC 539.3:517.958 L. V. BaevA problem of propagation of longitudinal and transverse waves in a multimodulus elastic isotropic medium is considered. In the model used, the medium is described by a potential depending on three invariants of strains, which allows the influence of preliminary deformation of the medium on the longitudinal and transverse velocities to be taken into account. 1.In considering adiabatic processes of elastic deformation, it is usually postulated that internal energy depends on invariants of strain measures. In the general form, the tensor relation between the stresses and strains for a nonlinearly elastic isotropic medium can be written asHere K 0 , K 1 , and K 2 are the functions of three invariants of the strain tensor I 1 = ε mm , I 2 = ε mn ε nm , and I 3 = ε mn ε np ε pm (summation is performed over repeated indices from 1 to 3).If a potential W = W (I 1 , I 2 , I 3 ) exists, we obtainOne of the methods most frequently used in the classical elasticity theory is the choice of the potential in the form of the expansion of internal energy into a Taylor series with respect to the initial state with retaining a certain number of terms. If the second, third, and fourth powers of strains are retained, we have nine terms. The constants at the corresponding powers of strains are called the Lame constants of the second, third, and fourth order. Other notations for the potential can also be used, for instance, those in Landau and Murnaghan nonlinear models. The relations between the constants in these models and the values of some constants can be found in [1]. These models can also be used to describe the behavior of materials with different resistances to tension and compression. It is assumed that these moduli can become different, in particular, owing to internal damages of the material and appearance of cracks, which makes the elastic characteristics and the character of propagation of elastic waves in the material depend on the type of the stress state. The qualitative characteristic responsible for the type of the strain state is the third invariant of the strain tensor, while the first and second invariants characterize the changes in the volume and shape. The difference in resistances to tension and compression can be taken into account to a certain extent by introducing a term depending on ξ = I 1 √ I 2 into the potential. Potentials of this type were considered in [2,3]. The potential in [3] was chosen in the form W (ε i,j ) = 0.5[K + ϕ(ξ)]I 2 1 + [G + ψ(ξ)](I 2 − I 2 1 /3), where the functions ϕ and ψ are related as 3ξ 2 ϕ +2(3−ξ 2 )ψ = 0. In this case, however, experiments on tension and compression with measurements of longitudinal and transverse strains yield four relations for determining a smaller Lavrent'ev Institute