TWMS J. Pure Appl. Math. V.12, N.2, 2021, pp.223-242 DYNAMICS OF THE MOVING RING-LOAD ACTING IN THE INTERIOR OF THE BI-LAYERED HOLLOW CYLINDER WITH IMPERFECT CONTACT BETWEEN THE LAYERS S.D. AKBAROV 1,2 , M.A. MEHDIYEV 2 , A.M. ZEYNALOV 3 Abstract. The dynamics of the moving-with-constant-velocity internal ring-load (-pressure) acting on the inner surface of the bi-layered hollow circular cylinder is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linear theory of elastodynamics. It is assumed that the internal pressure is point-located with respect to the cylinder axis and is axisymmetric in the circumferential direction. Moreover, it is assumed that shear-spring type imperfect contact conditions on the interface between the layers of the cylinder are satisfied. The focus is on determination of the critical velocity with analyses of the interface stress distribution and their attenuation rules with respect to time. At the same time, there is analysis of the problem parameters such as the ratio of modulus of elasticity, the ratio of the cylinder’s layers thickness to the external radius of inner layer-cylinder, and the shear-spring type parameter which characterizes the degree of the contact imperfection on the values of the critical velocity and stress distribution. Corresponding numerical results are presented and discussed. In particular, it is established that the values of the critical velocity of the moving ring-load increase with the thickness of the external layer of the cylinder. Keywords: moving ring-load, critical velocity, bi-layered hollow cylinder, shear-spring type im- perfection, interface stresses. AMS Subject Classification: 74F10. 1. Introduction A cannon bullet has its initial velocity through the moving in the interior of the cannon which in many cases can be modelled as a circular hollow cylinder. The moving of the bullet accompanied by a certain internal pressure which can be modelled as a moving axisymmetric ring-load. In order to control the safety and to modernize of the cannons it is necessary to study and to know the patterns of the dynamics of the aforementioned moving internal pressure. As the main issue in the investigations related to the moving load is to determination of the critical velocity of the moving load under which the resonance type accident takes place, therefore the aforementioned studies must be focused on the determination of this critical velocity and on the influence of the problem parameters on this velocity. In the sense of the modernization of cannons, for instance, it is interesting to know how the use of the bi-layered cylinder instead of the homogeneous hollow cylinder can act on the values of the critical velocity of the moving internal pressure and how it can be controlled the values of this critical velocity through the mechanical and geometrical parameters of the additional external hollow cylinder. Namely, the study of these and other related questions is the subject of the present paper. 1 Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey 2 Azerbaijan State University of Economics, Baku, Azerbaijan 3 Ganja State University, Ganja, Azerbaijan e-mail: [email protected], [email protected]Manuscript received April 2019. 223
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TWMS J. Pure Appl. Math. V.12, N.2, 2021, pp.223-242
DYNAMICS OF THE MOVING RING-LOAD ACTING IN THE INTERIOR
OF THE BI-LAYERED HOLLOW CYLINDER WITH IMPERFECT
CONTACT BETWEEN THE LAYERS
S.D. AKBAROV1,2, M.A. MEHDIYEV2, A.M. ZEYNALOV3
Abstract. The dynamics of the moving-with-constant-velocity internal ring-load (-pressure)
acting on the inner surface of the bi-layered hollow circular cylinder is studied within the scope
of the piecewise homogeneous body model by employing the exact field equations of the linear
theory of elastodynamics. It is assumed that the internal pressure is point-located with respect
to the cylinder axis and is axisymmetric in the circumferential direction. Moreover, it is assumed
that shear-spring type imperfect contact conditions on the interface between the layers of the
cylinder are satisfied. The focus is on determination of the critical velocity with analyses of
the interface stress distribution and their attenuation rules with respect to time. At the same
time, there is analysis of the problem parameters such as the ratio of modulus of elasticity,
the ratio of the cylinder’s layers thickness to the external radius of inner layer-cylinder, and
the shear-spring type parameter which characterizes the degree of the contact imperfection on
the values of the critical velocity and stress distribution. Corresponding numerical results are
presented and discussed. In particular, it is established that the values of the critical velocity
of the moving ring-load increase with the thickness of the external layer of the cylinder.
A cannon bullet has its initial velocity through the moving in the interior of the cannon
which in many cases can be modelled as a circular hollow cylinder. The moving of the bullet
accompanied by a certain internal pressure which can be modelled as a moving axisymmetric
ring-load. In order to control the safety and to modernize of the cannons it is necessary to study
and to know the patterns of the dynamics of the aforementioned moving internal pressure. As
the main issue in the investigations related to the moving load is to determination of the critical
velocity of the moving load under which the resonance type accident takes place, therefore the
aforementioned studies must be focused on the determination of this critical velocity and on
the influence of the problem parameters on this velocity. In the sense of the modernization of
cannons, for instance, it is interesting to know how the use of the bi-layered cylinder instead
of the homogeneous hollow cylinder can act on the values of the critical velocity of the moving
internal pressure and how it can be controlled the values of this critical velocity through the
mechanical and geometrical parameters of the additional external hollow cylinder. Namely, the
study of these and other related questions is the subject of the present paper.
1Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey2Azerbaijan State University of Economics, Baku, Azerbaijan3Ganja State University, Ganja, Azerbaijan
To determine the significance and place of the present investigations among the other related
ones we consider a brief review of those and note that the first attempt in this field was made
by Achenbach et al. [2] in which the dynamic response of the system consisting of the covering
layer and half plane to a moving load was investigated with the use of the Timoshenko theory
for describing the motion of the plate. However, the motion of the half-plane was described by
using the exact equations of the theory of linear elastodynamics and the plane-strain state was
considered. It was established that critical velocity exists in the cases where the plate material
is stiffer than that of the half-plane material. Reviews of later investigations, which can be taken
as developments of those started in the paper [2], are described in the papers [21, 28]. At the
same time, in the paper [21] the critical velocity of a point-located time-harmonic varying and
unidirectional moving load which acts on the free face plane of the plate resting on the rigid
foundation, was investigated. The investigations were made within the scope of the 3D exact
equations of the linear theory of elastodynamics and it was established that as a result of the
time-harmonic variation of the moving load, two types of critical velocities appear: the first
(the second) of which is lower (higher) than the Rayleigh wave velocity in the plate material.
Later, similar results were also obtained in the papers [7-10], which were also discussed in the
monograph [5]
Note that up to now, a certain number of investigations have also been made related to the
dynamics of the moving load acting on the pre-stressed system. For instance, in the paper
[27], the initial stresses on the values of the critical velocity of the moving load acting on an
ice plate resting on water were taken into account. In this paper, the motion of the plate is
described by employing the Kirchhoff plate theory and it is established that the initial stretching
(compression) of the plate along the load moving direction causes an increase (a decrease) in
the values of the critical velocity.
The paper [28] deals with the investigation of the lateral vibration of the beam on an elastic
half-space due to a moving lateral time-harmonic load acting on the beam, the initial axial
compression of this beam is also taken into consideration. In this investigation, the motion
of the beam is written through the Euler-Bernoulli beam theory, however, the motion of the
half-space is described by the 3D exact equations of elastodynamics and it is assumed there are
no initial stresses in the half-space.
Moreover, the influence of the initial stresses acting in the half-plane on the critical velocity of
the moving load which acts on the plate which covers this half-plane was studied in the papers
[16-19] in which the plane strain state was considered and the motion of the half-plane was
written within the scope of the three-dimensional linearized theory of elastic waves in initially
stressed bodies. However, the motion of the covering layer, which does not have any initial
stresses, as in the paper [2], was written by employing the Timoshenko plate theory and the
plane-strain state is considered.
Moreover, in the paper [6], the influence of the initial stresses in the covering layer and half-
plane on the critical velocity of the moving load acting on the plate covering the half-plane,
was studied. The same moving load problem for the system consisting of the covering layer,
substrate and half-plane was studied in the paper [22]. The dynamics of the system consisting
of the orthotropic covering layer and orthotropic half plane under action of the moving and
oscillating moving load were investigated in the papers [8, 9] and the influence of the initial
stresses in the constituents of this system on the values of the critical velocity was examined in
the paper [24].
S.D. AKBAROV, et al.: DYNAMICS OF THE MOVING RING-LOAD ACTING ... 225
Finally, we note the paper [11] in which the dynamics of the lineally-located moving load acting
on the hydro-elastic system consisting of elastic plate, compressible viscous fluid and rigid wall
were studied. It was established that there exist cases under which the critical velocities appear.
Note that related non-linear problems regarding the moving and interaction of two solitary
waves was examined in the paper [20] and other ones cited therein. At the same time, non-
stationary dynamic problems for viscoelastic and elastic mediums were studied in the papers
[25, 3] and other works cited in these papers. Moreover, it should be noted that up to now it
has been made considerable number investigations on fluid flow-moving around the stagnation-
point (see the paper [14] and other works listed therein). Some problems related to the effects of
magnetic field and the free stream flow-moving of incompressible viscous fluid passing through
magnetized vertical plate were studied in [16].
The investigations related to the fluid-plate and fluid-moving plate interaction were considered
the papers [12, 13].
It follows from the foregoing brief review that almost all investigations on the dynamics of
the moving and oscillating moving load relate to flat-layered systems. However, as noted in
the beginning of this section, cases often occur in practice in which it is necessary to apply a
model consisting of cylindrical layered systems, one of which is the hollow cylinder surrounded
by an infinite or finite deformable medium under investigation of the dynamics of a moving or
oscillating moving load. It should be noted that up to now some investigations have already been
made in this field. For instance, in the paper by Abdulkadirov [1] and others listed therein, the
low-frequency resonance axisymmetric longitudinal waves in a cylindrical layer surrounded by
an infinite elastic medium were investigated. Note that under “resonance waves” the cases under
which the relation dc/dk = 0 occurs, is understood, where c is the wave propagation velocity and
k is the wavenumber. It is evident that the velocity of these “resonance waves” is the critical
velocity of the corresponding moving load. Some numerical examples of “resonance waves” are
presented and discussed. It should be noted that in the paper [1], in obtaining dispersion curves,
the motion of the hollow cylinder and surrounding elastic medium is described through the exact
equations of elastodynamics, however, under investigation of the displacements and strains in
the hollow cylinder under the wave process, the motion of the hollow cylinder is described by
the classical Kirchhoff-Love theory.
Another example of the investigations related to the problems of the moving load acting on the
cylindrical layered system is the investigation carried out in the paper [33] in which the critical
velocity of the moving internal pressure acting in the sandwich shell was studied. Under this
investigation, two types of approaches were used, the first of which is based on first order refined
sandwich shell theories, while the second approach is based on the exact equations of linear
elastodynamics for orthotropic bodies with effective mechanical constants, the values of which
are determined by the well-known expressions through the values of the mechanical constants
and volumetric fraction of each layer of the sandwich shell. Numerical results on the critical
velocity obtained within these approaches are presented and discussed. Comparison of the
corresponding results obtained by these approaches shows that they are sufficiently close to each
other for the low wavenumber cases, however, the difference between these results increases with
the wavenumber and becomes so great that it appears necessary to determine which approach
is more accurate. It is evident that for this determination it is necessary to investigate these
problems by employing the exact field equations of elastodynamics within the scope of the
piecewise homogeneous body model, which is also used in the present paper. The other drawback
of the approach used in the paper [33] is the impossibility to calculation of the interface stresses
between the cylinder’s layers with enough accuracy. It is evident that such calculation of the
226 TWMS J. PURE APPL. MATH., V.12, N.2, 2021
interface stresses from which depends directly the adhesion strength of the layered cylinder,
must be also made within the scope of the piece-wise homogeneous body model with employing
the exact equations and relations of the elastodynamics.
Taking the foregoing statements into account, in the present paper we attempt to investigate
the dynamics of the moving ring load in the interior of the bi-layered hollow cylinder within
the scope of the piecewise homogeneous body model with the use of the exact equations and
relations of the elastodynamics. During this investigation the main attention is focused not only
on the critical velocity of the moving load and on the influence of the problem parameters on
this velocity, but also on the interface stresses appearing as a result of the mentioned moving
load.
2. Mathematical formulation of the problem
Consider a bi-layered hollow circular cylinder with internal (external) layer thickness h2 (h1)
(Fig.1). We associate the cylindrical and Cartesian systems of coordinates Orθz and Ox1x2x3(Fig. 1) with the central axis of this cylinder. Assume that the external radius of the cross section
of the inner layer-cylinder is R and on its inner surface axisymmetric uniformly distributed
normal ring-forces moving with constant velocity V act and these forces are point-located with
respect to the cylinder central axis. Within this framework, we investigate the axisymmetric
stress-strain state in this bi-layered cylinder by employing the exact equations of the linear
theory of elastodynamics within the scope of the piecewise homogeneous body model. Below,
the values related to the inner and to the outer layer will be denoted by upper indices (2) and
(1), respectively.
Figure 1. The geometry of the system under consideration.
We suppose that the materials of the constituents are homogeneous and isotropic. We write
the field equations and contact conditions as follows.