Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 2 2000
Geom., Cont. and Micros., II
A.V. Porubov
∗
STRAIN SOLITARY WAVES IN AN ELASTIC ROD WITH MICROSTRUCTURE
Abstract. The nonlinear longitudinal strain solitary waves are studied inside cylin- drical elastic rod with microstructure. The problem is solved using the pseudo-
continuum Cosserat model and the Le Roux continuum model. A procedure is developed for derivation of a governing equation for longitudinal nonlinear strain
waves. Exact solution of the equation has the form of a travelling bell-shaped soli- tary wave. The influence of microstructure on the solitary wave propagation is
studied. Possible experimental determination of the parameters of the microstruc- ture is discussed.
1. Introduction
Sometimes classic elastic theory cannot account for phenomenon caused by the microstructure of a material. A particular case is a dispersion of strain waves in an elastic medium. The influ-
ence of microstructure may provide dissipative effects [14, 6, 2], however, here consideration is restricted by non-dissipative case. The theory of microstructure has been developed recently,
see [6, 7, 15, 17] and references therein. Most of results belong to the linear theory of elasticity, however, there are findings in the field of the nonlinear theory [6, 7]. Strain waves were studied
mainly in the linear approximation [7, 15, 17]. Only a few works are devoted to the nonlinear waves in microstructured non-dissipative media [6, 19, 20, 10, 9]. Waves in elastic wave-guides
with microstructure were out of considerable investigation. Also the values of the parameters characterizing microstructure, are unknown as a rule, only a few data may be mentioned [20].
It is known that the balance between nonlinearity and dispersion may result in an appear- ance of bulk localized long bell-shaped strain waves of permanent form solitary waves or soli-
tons which may propagate and transfer energy over the long distance along an elastic wave guide. The amplification of them may cause the appearance of plasticity zones or microcracks
in a wave guide. This is of importance for an assessment of durability of elastic materials and structures, methods of nondestructive testing, determination of the physical properties of elastic
materials, particularly, polymeric solids, and ceramics. Bulk waves provide better suited de- tection requirements than surface strain waves in setting up a valuable nondestructive test for
pipelines.
Recently, the theory has been developed to account for long longitudinal strain solitary waves propagating in a free lateral surface elastic rod [5, 21, 22]. The procedure has been pro-
posed to obtain model equations using boundary conditions on the rod surface [18]. The nonlin- earity, caused by both the finite stress values and elastic material properties, and the dispersion
resulting from the finite transverse size of the rod, when in balance allow the propagation of
∗
This research has been supported by the INTAS under Grant 99-0167.
189
190 A.V. Porubov
strain solitary waves v. The equation governing this process is of Boussinesq type, namely, a double dispersive equation
v
t t
− α
1
v
x x
− α
2
v
2 x x
+ α
3
v
x x t t
− α
4
v
x x x x
= 0.
The coefficients α
i
depend upon the elastic parameters of the rod material. Exact solution of the equation has the form of a travelling bell-shaped solitary wave. The amplitude and the velocity
of the solitary wave are explicitly connected with the elastic moduli. It allows to propose the estimation of the Murnaghan third order elastic moduli using measurement of the solitary wave
parameters [1]. Motivated by analytical theoretical predictions, there has been successful exper- imental generation of strain solitons in a polystyrene free lateral surface rod using holographic
interferometry [3]. The procedure developed in Ref.[18], has been successfully applied for the more complicated modelling of strain waves in a narrowing rod[4] and in a rod interacting with
an another external elastic medium [1].
The present paper refers to the study of nonlinear solitary waves inside cylindrical rod with microstructure. The problem is solved using the ”pseudocontinuum” Cosserat model and the
Le Roux continuum model. A procedure is developed for derivation of the model equation for long longitudinal strain waves inside the rod. The influence of the microstructure on the solitary
wave propagation is studied. Possible experimental determination of the parameters of the microstructure is discussed.
2. Modelling of elastic medium with microstructure
