Shock and Vibration
Volume 2017 (2017), Article ID 5652023, 12 pages
https://doi.org/10.1155/2017/5652023

Research Article

Analysis of the Viscoelastic Sphere Impact against a Viscoelastic Uflyand-Mindlin Plate considering the Extension of Its Middle Surface

Yury A. Rossikhin, Marina V. Shitikova, and Phan Thanh Trung

Research Center for Wave Dynamics of Solids and Structures, Voronezh State Technical University, 20 Letija Oktjabrja Str. 84, Voronezh 394006, Russia

Correspondence should be addressed to Marina V. Shitikova; shitikova@vmail.ru

Received 7 August 2016; Accepted 13 November 2016; Published 17 January 2017

Academic Editor: Ivo Caliò

Copyright © 2017 Yury A. Rossikhin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the present paper, the problem on impact of a viscoelastic sphere against a viscoelastic plate is considered with due account for the extension of plate’s middle surface and local bearing of sphere and plate’s materials via the Hertz theory. The standard linear solid models with conventional derivatives and with fractional-order derivatives are used as viscoelastic models, respectively, outside and within the contact domain. As a result of impact, transient waves (surfaces of strong discontinuity) are generated in the plate, behind the wave fronts of which up to the boundaries of the contact domain the solution is constructed in terms of one-term ray expansions due to short-time duration of the impact process. The motion of the contact zone occurs under the action of extension forces acting in the plate’s middle surface, transverse force, and the Hertzian contact force. The suggested approach allows one to find the time-dependence of the impactor’s indentation into the target and the Hertzian contact force.

1. Introduction

Nowadays fractional calculus is widely used in different fields of science and technology [1–11], including various dynamic problems of mechanics of solids and structures [12]. The problems of impact response of thin-walled structures belong to the most challenging ones among the dynamic problems. The review of papers dealing with the application of fractional calculus models for solving the impact problems could be found in [12, 13].

However, to the authors’ knowledge, the impact response of viscoelastic plates utilizing the fractional derivative models has been treated only in a few articles. Thus, the problems of impact of an elastic rod or sphere against a viscoelastic Kirchhoff-Love plate or against a viscoelastic Uflyand-Mindlin plate have been considered in [14] and [15–17], respectively. In the case of the classical plate, the method of the Green function construction was used as the method of solution [14]. In the case of the Uflyand-Mindlin plate, the ray method was utilized, according to which the solution behind the fronts of transient waves (surfaces of strong discontinuity) is constructed in terms of truncated power series with variable coefficients. Moreover, the contact force has been taken into account either according to the Hertz contact theory [15] or in terms of linear approximations allowing one to apply Laplace transform technique [16, 17].

In all enumerated above papers, the extension of the plate’s middle surface, as well as viscoelastic properties of an impactor, has not been considered. However, as experimental research of the behaviour of flexural thin-walled structures under low and middle velocity impacts shows [18], the contribution of extensional deformations into the impact process is rather essential and, therefore, it should be taken into account. Thus, the impact of elastic bodies upon elastic beams and plates with consideration for the transverse deformations and extension of a middle surface was studied in [19], and recently this approach has been considered when investigating the dynamic response of a viscoelastic Timoshenko-type beam impacted by an elastic sphere [20].

In the present paper, we will generalize the approaches developed recently in [20, 21] when investigating the dynamic response of a viscoelastic beam impacted by an elastic sphere with due account for beam’s middle surface extension and by a viscoelastic sphere ignoring the middle surface extension, respectively, over the problem of theoretical analysis of the dynamic response of the viscoelastic plate by considering the influence of the middle surface extension on the process of deformation of the viscoelastic plate during the impact by a viscoelastic spherical impactor.

In both papers [20, 21], the viscoelastic properties of the beam have been described by the standard linear solid model with integer time derivatives. During the impact process there occurs decrosslinking within the domain of the contact of the beam with the impactor, resulting in more freely displacements of molecules with respect to each other, and finely in the decrease of the beam material viscosity in the contact zone. This circumstance allows one to describe the behaviour of the beam material within the contact domain by the standard linear solid model involving fractional derivatives, since variation in the fractional parameter (the order of the fractional derivative) enables one to control the viscosity of the beam material.

It is known that viscoelastic properties of the material could be changed during its life-cycle due to different external effects, among them aging and impact loads [22], and/or temperature and irradiation [23]. Dynamic mechanical studies of irradiated polyethylene were presented by Merrill et al. [23], wherein it was found the influence of irradiation on viscoelastic characteristics of polyethylene. Experimental study of concrete aging effect on the contact force and contact time during impact interaction of an elastic rod with a viscoelastic beam was carried out by Popov et al. [22], and a good correlation with the analytical model proposed in [24] was declared.

In the present paper, following [20, 21, 24] we will study the dynamic response of a viscoelastic Uflyand-Mindlin plate impacted by a viscoelastic sphere considering the extension of plate’s middle surface. The standard linear solid model with conventional derivatives and with fractional-order derivatives will be used as viscoelastic models, respectively, outside and within the contact domain. As a result of impact, transient waves (surfaces of strong discontinuity) are generated in the plate, behind the wave fronts of which up to the boundaries of the contact domain the solution is constructed in terms of one-term ray expansions due to short-time duration of the impact process. The motion of the contact zone is analyzed under the action of extension forces acting in the plate’s middle surface, transverse forces, and the Hertzian contact force using the algebra of Rabotnov’s fractional operators.

2. Problem Formulation and Governing Equations

Let a viscoelastic sphere of radius and mass move with the constant velocity towards a viscoelastic plate along the -axis, which is perpendicular to the plane of the plate. Impact occurs at . As a result of transverse impact against the viscoelastic plate, transient waves, the longitudinal wave, and the wave of transverse shear propagate along the plate, the fronts of which are the surfaces of strong discontinuity. These waves are cylindrical surfaces-strips, the generators of which are parallel to the normal to the median plane, while the guides located in the middle plane are the circumferences extending with the normal velocities . The index indicates the ordinal number of the wave: for the longitudinal wave and for the transverse wave.

A certain desired function behind the fronts of wave surfaces and could be represented in terms of the ray series [25]

where are discontinuities in the th-order time-derivatives of the desired function on the waves surfaces ; that is, at , is the polar radius, the upper indices “+” and “−” denote that the values are calculated immediately ahead of and behind the wave fronts, respectively, and is the unit Heaviside function.

Since the impact process is of short duration, then it is possible to restrict only by zero-order terms, that is,

In order to find the discontinuities in the desired values, first it is necessary to write the governing equations describing the dynamic behaviour of the viscoelastic isotropic Uflyand-Mindlin plate in the polar coordinates

where and are the forces acting in the plate plane in the - and -directions, respectively, and are the bending moments, is the transverse force, is the plate deflection, is the displacement along the radius, is the rotational angle of the normal in the -direction, is the density, is the plate thickness, and are shear and Young’s operators, respectively, is the Poisson’s operator, is the operator of the cylindrical rigidity, is the shear coefficient, and overdots denote the time derivatives, and then to apply the kinematic condition of compatibility proposed in [26] for physical components of thin bodies

to (3) and (5), where is the Thomas -derivative [27] of the function preassigned on the moving surface.

Note that condition (11) is valid in the case when spatial coordinates coincide with the ray coordinates [28].

In order to simplify the procedure for utilizing condition (11), let us interpret the surface of strong discontinuity as a layer of small thickness , within which the desired value changes monotonically and continuously from the magnitude to the magnitude . Suppose that the ahead and back fronts of the shock layer arrive at a certain point with the fixed radius at the time instants and , respectively, where is small. Inside the shock layer the following relationship

is fulfilled.

Writing (3) and (5) within the shock layer and considering (12), we obtain

where , and .

Thus we have chosen only those equations of motions which will be utilized in further treatment.

Integrating (13) and (14) over from to and then tending yield

Relationships (15) and (16) are called the dynamic conditions of compatibility. During the deduction of these conditions we neglect the terms and in (3) and in (5), as well as the term in (12), since the integrals from these terms over from to tend to zero as .

In order to write (6) and (10) in discontinuities, it is necessary to decode the operators entering in these equations.

3. Decoding of Operators

For decoding the operators involved in (6) and (10), it is a need to specify two any operators, for example, the operator of volume extension-compression and shear operator . As numerous experiments with volume stresses and strains show [29], for the majority of materials operator is a constant value, that is,

where is a certain constant.

In further treatment the standard linear solid model with conventional derivatives and with fractional-order derivatives will be used as viscoelastic models, respectively, outside and within the contact domain, because during the impact process there occurs decrosslinking within the domain of the contact of the plate with the impactor, resulting in more freely displacements of molecules with respect to each other and finely in the decrease of the plate material viscosity in the contact zone. This circumstance allows us to describe the behaviour of the plate material within the contact domain by the standard linear solid model involving fractional derivatives, since variation in the fractional parameter (the order of the fractional derivative) enables one to control the viscosity of the plate material.

Thus for convenience of presentation, first we will decode fractional-order operators valid inside the contact domain, and then tending fractional parameter to unit in the final relationships we will obtain the corresponding formulae for the viscoelastic plate outside the contact domain in terms of the conventional viscoelasticity.

So within the contact domain of the impactor with the viscoelastic target, we would preassign this connectedness between the deviators of stress and strain by the standard linear solid model involving fractional derivatives [12], that is,

or

where and are the relaxation and retardation times, respectively,    is the fractional parameter, is the relaxed shear modulus,

is the Riemann-Liouville fractional derivative of the -order [12], is the Gamma-function, and is a certain function.

From (19) it follows that operator is given in the form

or

Here the operator

has been introduced, which has the name of the dimensionless Rabotnov fractional operator [13].

If we consider formula

which is proved by the direct check-up [13], in relationship (22) and introduce the following notation

where is the nonrelaxed shear modulus, then as a result we obtain

where is the dimensionless defect of modulus.

Now let us calculate the Young operator according to the formula

We will start with the operator

where .

We will seek the inverse operator in the form

where and are yet unknown constants.

Further we multiply operators (28) and (29), consider their property , and utilize the formula

which is proved by the direct check-up [13].

As a result we obtain

Equating to zero each bracket in (31), we are led to a set of two equations for defining and

whence it follows

Finally we substitute operators (26) and (29) in (27) and consider relationship (30). As a result we have

Considering that according to the set of (32) the first bracket in (34) is equal to zero, while the second bracket is as follows:

and we finally obtain

where , and .

Now let us calculate the compliance operator which is defined in the form

where and are yet unknown constants. To determine them, we will utilize the property and consider relationship (30). Then we obtain

Equating to zero each bracket in (38) yields

Solving the set of (39), we find

Now we could calculate the Poisson’s operator according to formula (17), which could be rewritten in the form

Considering (36), from formula (41) we have

For further treatment we should know the following operators:

In order to calculate the operators in the right-hand side of (43) and (44), we assume that they have the following form:

where , and , are yet unknown constants.

Equating the right sides of relationships (43), (45) and (44), (46), reducing the obtained expressions to the common denominator, and considering (30) yield

Vanishing to zero the expressions in square brackets in (47), we determine unknown constants

Now we could calculate the operator

For this purpose, we substitute (36), (45), and (46) in (49) with due account for (30). As a result we obtain

where

Finally we would calculate operator considering formulas (30) and (48)

Further for the convenience we will represent the Rabotnov fractional operator in another form multiplying the numerator and denominator of (23) by , where

is the fractional integral.

As a result we obtain

It has been shown in [13] that interpreting the right-hand side part of (54) as a sum of the infinite descending geometrical progression with the denominator , we have

or

where

is Rabotnov’s fractional exponential function [29, 30], which at goes over into the conventional exponent, that is,

All formulas which have been obtained for the viscoelastic fractional derivative standard linear solid model remain valid for the standard linear solid model with conventional derivatives.

Since out of the contact domain viscoelastic features of plate’s material are described via the standard linear solid model with conventional derivatives, then it is possible to rewrite (6) and (10) with due account for the derived above operators putting . As a result we have

In order to write relationships (59) in discontinuities, let us substitute there and by and , respectively, according to (12), and then write the obtained relationships at the moments and . Thus,

Expanding the integrals entering in (61) and (63) in Taylor series in terms of the small parameter and limiting ourselves by the zero and first approximations yield

where may coincide with , , or , and .

Subtracting (61) from (60) and (63) from (62), considering relationship (64) as well as that , , , , , , and tending to zero, we have

From expressions (15) and (65), (16), and (66) it is possible to find velocities of two types of transient waves, namely, longitudinal-flexural wave and transverse shear wave:

Substituting the found velocities (67) and (68) in formulas (15) and (16), and limiting, as it has been already mentioned, by the zeroth terms of the ray expansions, we obtain

Relationships (69) and (70) by their form do not differ from the corresponding relationships for an elastic plate, with the only difference that a subindex has appeared at the velocity notation, which indicates that the velocities are expressed in terms of the nonrelaxed modulus of the viscoelastic model (see Appendix).

4. Equations of Motion of the Contact Domain and the Impactor

At the displacement of the sphere’s center could be represented in the form

where is the impactor’s indentation due to the local bearing of target and impactors’ materials within the contact domain.

Then the equation of motion of the part of the plate being in contact with the sphere and the equation of the sphere have the form

where is the radius of the contact domain, and is the contact force.

Equations (72) and (73) could be solved with due account for formulas (69) and (70), as well as considering the following initial conditions:

The relationships

following from the generalized contact Hertz theory should be added to (72) and (73), where

and in doing so the indices “1” and “2” refer to the viscoelastic plate and viscoelastic sphere, respectively. The operators valid within the contact domain are based on the fractional derivative standard linear solid models.

Integrating (73) yields

In order to express the contact force in terms of displacement , it is needed to decode the operator . The procedure for decoding this operator as well as the construction of its inverse operator has been recently described in detail in Rossikhin et al. [21]. Thus, it has been shown [21] that operator could be written in the form

where

and unknown constants and could be determined from the set of equations

wherein

Considering (79), the contact force is defined as

where .

Utilizing (83), it is possible to rewrite (78) in the form

The values of and defined by (69) and (70), respectively, should be substituted in (72); then this equation together with (84) will give us a set of two governing equations involving only two unknown values, and . However, the force entering in (72) depends on the velocity according to (69); thus it is necessary to express it in terms of and .

With this purpose in mind, we write the stress tensor for a three-dimensional viscoelastic medium

where are components of the displacement vector, summation is carried out over two repeated indices, and is the Kronecker symbol, , an index after a comma labels the derivative with respect to the coordinates , , and .

Considering formula (26), as well as the expression for operator

where , which follows from the formula

let us rewrite (85) in the form

Using the procedure applied above for deducing (65) and (66), from (88) we obtain

Applying the geometric condition of compatibility

where are the components of the unit vector normal to the wave surface, and are the components of the unit vector normal to the plate, to (89), yields

where , and .

Note that condition (90) differs from the Hadamard condition of compatibility, since (90) involves the second term taking the transverse shear deformation of the plate into account.

Multiplying (91) by , we find

Considering that for the cylindrical wave-strip propagating in the plate and in the cylindrical set of coordinates and that , we are led to the following relationship:

wherein the value , in its turn, is defined by the formula

Then the discontinuity could be written as

and the value due to (69) takes the form

Considering formulas (70), (76), (83), and (96), as well as the relationship

which is obtained from formula (12), if we substitute there the function by the function , we could rewrite (72) in the form

where

Note that since the impact process is of short duration, then in the integrals entering in (84) and (98)

Consider (100) and (84) and (98) are reduced to

Adding formula (71) to (101) and (102), we obtain the closed set of three equations in three unknown functions, namely, , , and .

5. Approximate Analytical Solution of the Governing Set of Equations

Now we will consider several particular cases.

5.1. The Case

Let us consider first the case when which corresponds to an elastic target and elastic impactor.

Since at

as it has been shown in [21], then in (101) and (102) the terms involving the sums vanish to zero; therefore, (102) is reduced to

while (101) could be retained in the form of (73).

If we neglect the inertia of the contact domain due to its smallness, then it could be rewritten as

Consider relationships

and introduce a new variable

Equation (73) with due account of (105) could be rewritten in the form

where .

If we ignore the deformation of the middle surface, that is, putting , then (108) is reduced to

where .

Note that (109) is the Abel equation of the second kind [31], and its solution was found in [25, 32] in the form

where

It has been emphasized in [25, 32] that coefficients and define two different processes being caused by the shock interaction of the impactor and the target: the coefficient is responsible for the dynamic processes generating in the plate during propagation of the surfaces of the discontinuity, while the coefficient describes the quasistatic processes occurring at the local bearing of the material due to Hertzian theory. If , what corresponds to an infinite large velocity of transverse shear wave propagation (Kirchhoff-Love plate), then solution (110) for small goes over into the quasistatic solution [18].

If in (108), then we are led to the well-known equation which describes the impact interaction of colliding elastic bodies according to the Hertz theory [33].

If , then since is a small value, (108) could be rewritten in the form

Let us put in the right-hand side of (112) and integrate it. As a result we obtain

From (113) we determine the refined duration of contact

where , as well as the refined time at which attains its maximal value

where , and the maximal indentation

where .

5.2. The Case

At , (101) and (102) after eliminating the inertia of the contact domain due to its smallness and considering that take the form

where , or after eliminating the value from (117)

From (118) we find the refined contact duration

as well as the refined time at which attains its maximal value

and the maximal indentation

Comparing formulas (114) and (119), (115) and (120), as well as (116) and (121), the following conclusion could be made: all characteristic values calculated at are larger than those calculated at , and all characteristic values calculated at other magnitudes of    locate between these two corresponding limiting values in order of increasing magnitudes of the value .

Appendix

Let us consider for simplicity the dynamic behaviour of a viscoelastic rod which is described by a set of equations

where is the stress, is the displacement, is the density, is the time, is the nonrelaxed magnitude of compliance, , is the creep function, and is the creep kernel.

If we apply the condition of compatibility (11) to (A.1) and (A.2) and utilize to (A.2) the procedure adopted in the present paper for the deduction of relationships (65) and (66), then as a result we obtain from (A.1) the Hadamard dynamic condition of compatibility

and from the rheological equation (A.2) the relationship

where is the nonrelaxed elastic modulus, and .

Eliminating the value from (A.3) and (A.4), we have

If one considers a pure elastic rod, then the same formulas (A.3)–(A.5) could be obtained, with the only difference that, for the elastic rod, as distinct from a viscoelastic one, there exists only one elastic modulus.

From the physical point of view, such a coincidence could be explained by the fact that discontinuities of the desired values on the front of the shock wave proceed very quickly; therefore on the wave front a viscoelastic material behaves itself as an elastic with the nonrelaxed modulus of elasticity.

Another matter if besides the interconnection between the discontinuities in stress and velocity it is necessary to find the time-dependence of the discontinuities and . In this case it is a need to differentiate (A.1) and (A.2) times and with respect to time , respectively, and utilize the compatibility condition (11) wherein is substituted by , resulting in the following condition of compatibility:

Eliminating then the values    from the equations obtained after such a procedure, we are led to the recurrent equations [34, 35]

Note that the discontinuity in the displacement is equal to zero, that is, , since the medium is considered to be continuous without any cracks or damage. In other words, the function could not be connected with the Heaviside function , as the Heaviside function assumes discontinuities, while the function is continuous and does not experience any discontinuities with time.

Integrating (A.7) yields

where

Constants are assigned on the rod’s boundary due to the condition

From (A.9) at we find the zero-order term of the ray series for the viscoelastic rod

while for the elastic rod we have

Equation (A.8) at is reduced to the dynamic condition of compatibility for the rod in the form of relationship (A.3).

In more intricate cases such as propagation of cylindrical wave-strips in viscoelastic plates, the discontinuities in desired values are proportional to , where , and is the initial radius of the nonstationary cylindrical wave-strip (see, e.g., [36]).

Since in the present paper there is no need to find the time-dependence of the zeroth-order discontinuities, then we have restricted ourselves by the deduction of the dynamic conditions of compatibility (15) and (16) connecting the discontinuities in the generalized forces with the discontinuities in the generalized velocities, and these formulae differ in nothing in regard to the form from those for an elastic plate.

Disclosure

Phan Thanh Trung is on leave from HCMC University of Technology and Education, Vietnam.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Ministry Education and Science of the Russian Federation.

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