# Impact response of a Timoshenko-type viscoelastic beam considering the extension of its middle surface

- Yury A. Rossikhin
^{1}, - Marina V. Shitikova
^{1}Email author and - Maria Guadalupe Estrada Meza
^{1}

**Received: **2 September 2015

**Accepted: **12 February 2016

**Published: **27 February 2016

## Abstract

In the present paper, the problem of low-velocity impact of an elastic sphere against a viscoelastic Timoshenko-type beam is studied considering the extension of its middle surface. The viscoelastic features of the beam out of the contact domain are governed by the standard linear solid model with derivatives of integer order, while within the contact domain the fractional derivative standard linear solid model is utilized, in so doing rheological constants of the material in both models are the same. However the presence of the additional parameter, i.e. fractional parameter which could vary from zero to unit, allows one to vary beam’s viscosity, since the structure of the beam’s material within this zone may be damaged, resulting in the decrease of the beam material viscosity in the contact zone. Consideration for transient waves (surfaces of strong discontinuity) propagating in the target out of the contact zone via the theory of discontinuities and determination of the desired values behind the surfaces of discontinuities upto the contact domain with the help of ray series, as well as the utilization of the Hertz theory in the contact zone allow one to obtain a set of two integro-differential equations, which govern the desired values, namely: the local bearing of the target and impactor’s materials and the displacement of the beam within the contact domain.

## Keywords

## Background

For the first time the extension of the middle surface during impact upon a thin body has been taken into account in the problems of impact of an elastic sphere upon an elastic Timoshenko beam in (Rossikhin and Shitikova 1996) and upon an elastic Timoshenko-type thin-walled beam of open profile in (Rossikhin and Shitikova 1999). In the state-of-the-art article (Rossikhin and Shitikova 2007a) devoted to the wave theory of impact and published in 2007 in “The Shock and Vibration Digest”, the impact response of an elastic Uflyand–Mindlin plane and a Timoshenko beam was analyzed in detail.

*v*] is the discontinuity in the velocity of longitudinal displacement on the plane transient wave propagating along the beam during the impact process with the velocity \(G_1\,=\,\sqrt{E/\rho }\),

*E*is Young’s modulus, \(\rho\) is the density, \(\nu\) is Poisson’s ratio, \(h_b\) is the beam thickness, and \(\alpha\) is the local bearing the target and impactor’s materials. The above formula is valid for an elastic thin plate, but it is invalid for a beam.

*N*is the longitudinal force,

*w*and

*Q*are transverse displacement and force, respectively,

*P*(

*t*) is the contact force,

*z*is the coordinate directed along the beam axis, a dot denotes the time-derivative,

*a*is the radius of the contact zone, in so doing the author of (Vershinin 2014) has considered that plane transient waves (surfaces of discontinuity) propagate from the contact zone during the process of impact. However, if the contact domain is a circular disk with a volume \(h\pi a^2\), then the waves travelling from a circular contact zone are diverging circles.

*aF*, where

*F*is the cross sectional area of the beam, and the equation should have the form

*K*is the shear coefficient, and \(\mu\) is the shear modulus. This equation is also incorrect.

Based on the aforesaid, the further numerical treatment presented in Vershinin (2014) occurs to be invalid.

In the present paper, we present not only the correct equation for the impact response of the elastic Timoshenko beam, but the deduction and analysis of equations describing the behaviour of a viscoelastic Timoshenko-type beam impacted by an elastic sphere are given considering the damage of the target material within the contact domain.

## Problem formulation

*R*and mass

*m*move along the \(y-\)axis with a constant velocity \(V_0\) towards a Timoshenko-type viscoelastic homogeneous isotropic beam of the width

*h*(Fig. 1), viscoelastic features of which are described by the standard linear solid model with conventional derivatives. The dynamic behaviour of such a beam with due account for extension of its middle surface is described by the following set of equations:

*M*,

*Q*, and

*N*are the bending moment, the shear and longitudinal forces, respectively,

*u*and

*w*are longitudinal and transverse displacements, respectively, \(\psi\) is the angle of rotation of the cross section around the

*z*-axis, \(v=\dot{u}\), \(W={\dot{w}}\), \(\Psi =\dot{\psi }\),

*F*and

*I*are the cross-sectional area and the moment of inertia with respect to the

*x*-axis, respectively, \(\rho\) is the density,

*K*is the shear coefficient dependent on beam’s geometrical dimensions and the form of its cross section, and an overdot denotes the time derivative.

*Z*(

*t*) is a desired function, \(E_\infty\) and \(E_0\) are the non-relaxed (instantaneous modulus of elasticity, or the glassy modulus) and relaxed elastic (prolonged modulus of elasticity, or the rubbery modulus) moduli which are connected with the relaxation time \(\tau _\varepsilon\) and retardation time \(\tau _\sigma\) by the following relationship:

*n*and \(t_\varepsilon\) are yet unknown constants.

*y*could be represented in the form

\(\Gamma [\gamma (n+1)]\) is the Gamma-function, \(\ni _\gamma \left( t/\tau _i \right)\) is Rabotnov’s fractional exponential function (Rabotnov 1948) which at \(\gamma =1\) goes over into the ordinary exponent, and operator \(\ni _\gamma ( \tau _i )\) transforms into operator \(\ni _1^* ( \tau _i )\). When \(\gamma \rightarrow 0\), the function \(\ni _\gamma \left( t/\tau _i \right)\) tends to the Dirac delta-function \(\delta (t)\).

This distinction is connected with the fact that 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 finally in the decrease of the beam material viscosity in the contact zone (Popov et al. 2015). 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 from its initial value at \(\gamma =1\) to its vanishing at \(\gamma =0\). Thus, the substitution of operators (5), (6), and (12) with operators (17)–(19), respectively, is quite reasonable.

*P*(

*t*) is defined by formula (15), while the equation of motion of the contact zone, which is considered to be rigid and is restricted by the planes \(z=\pm a\) (Fig. 1)

*Z*(

*z*,

*t*) behind the front of the wave surface is represented in terms of the ray series (Rossikhin and Shitikova 1995)

*k*-th order derivatives with respect to time

*t*of the desired function

*Z*(

*z*,

*t*) on the wave surface, the upper signs \(+\) and \(-\) denote that the given value is calculated immediately ahead of and behind the wave front, respectively, the index \(\alpha\) labels the ordinal number of the wave, namely: \(\alpha =1\) for the longitudinal wave, and \(\alpha =2\) for the transverse wave,

*H*(

*t*) is the Heaviside function, and \(G^{(\alpha )}\) is the normal velocity of the surface of discontinuity.

*k*times with respect to time, take their difference on the different sides of the wave surface \(\Sigma\) and apply the condition of compatibility for the \(k+1\)-order discontinuities of the function

*Z*, which has the following form (Rossikhin and Shitikova 1995):

*d*/

*dt*is the complete time-derivative of the function \(Z,_{(k)}(z,t)\) on the moving surface of discontinuity.

Since the process of impact is a transient process, then, firstly, it is possible to limit ourselves by zeroth terms of the ray series (28), and secondly, to neglect the waves reflected from the end face of the beam considering that they reach the contact zone after impactor’s rebound from the beam.

*M*with the coordinate

*z*at the moment of time

*t*, while the back front of the shock layer reaches this point at the moment \(t+\Delta t\). The desired values

*Z*(

*z*,

*t*) at the point

*M*, such as velocity, generalized forces and deformations, during the time increment \(\Delta t\) change monotonically and uninterruptedly from the magnitude \(Z^-\) to the magnitude \(Z^+\), in so doing within the layer, according to the condition of compatibility (28), the relationship

*t*to \(t+\Delta t\), and tending \(\Delta t\) to zero, we find

*t*and \(t+\Delta t\), we obtain

Note that relationships (43) differ nothing from those for an elastic beam, since at the moment of impact a viscoelastic medium behaves as an elastic medium with the unrelaxed elastic modulus.

Now it is necessary to substitute the values of *N* and *Q* defined by (43) in (25). However the governing set of two equations, (23) and (25), should involve only two unknown values, \(\alpha\) and *w*, while the force *N* entering in (25) depends on the velocity *v*, as it follows from (43), and therefore *v* should be expressed in terms of \(\alpha\) and *w*.

Note that formula (54) has been presented in Landau and Lifshitz (1965) for elastic beams.

*N*could be rewritten in the form

A solution of the set of two equations, involving differential equation (60) and integro-differential equation (62), allows one to find the time-dependence of the values *W* and \(\alpha\).

Equations (60) and (62) are subjected to the initial conditions (26).

### Solution of the problem in the case of neglecting the extension of the beam’s middle surface

*C*(

*t*) could be found using the method of variation of an arbitrary constant, resulting in

Considering (71), solution (75) for two limiting cases is reduced to the following:

### Solution of the problem in the case of considering the extension of the beam’s middle surface

#### Elastic target

*g*(\(\alpha\) is small) in (88), then it is reduced to

When \(g = 0\), which is realized at an infinitely large speed of the shear wave propagation, the solution (90) for small \(\alpha\) goes over into the quasi-static solution obtained by Timoshenko (1934) for the Bernoulli–Euler beam.

#### Viscoelastic target

*e*/

*g*governing the extension of the beam's middle surface.

#### Numerical example

For numerical analysis it is convenient to rewrite formulas (78)–(83) and (98)–(103) in the dimensionless form:

The dimensionless time \(t^*= t\left( t^0_\mathrm{max}\right) ^{-1}\) dependence of the dimensionless value \(\alpha ^*= \alpha \left( V_0 t^0_\mathrm{max}\right) ^{-1}\) characterizing the local bearing of impactor and target materials for different values of the fractional parameter \(\gamma\), which are indicated by figures near the corresponding curves, is shown in Fig. 2. Solid curves are calculated without considering the extension of the target middle surface, while dashed lines correspond to the case with due account for middle surface extension at \(\chi _1=2\) and \(\chi _2=0.5\).

## Conclusion

The main goal of this paper is to bring to light the physical sense of the fractional parameter in problems on impact, since one and the same question arises very often, namely: Why is it needed to introduce a fractional derivative in problems of mechanics? The authors have tried to answer this question at least for the problems of impact by connecting the fractional parameter with the changes in microstructure of beam’s material within the contact domain. For this purpose we have assumed that viscoelastic features of the beam outward the contact zone is determined by the standard linear solid with ordinary derivatives, while the contact force is also viscoelastic and its features are governed by the standard linear solid model with fractional derivatives, in so doing relaxed and non-relaxed moduli and relaxation and retardation times coincide with the corresponding moduli and times for the viscoelastic medium out of the contact zone, and the fractional parameter varies from zero till unit controlling the viscosity within the contact domain. This is connected with the fact that during the low-velocity impact there could occur decrosslinking within the domain of the contact of the beam with the sphere, resulting in more freely displacements of molecules with respect to each other, and finally in the decrease of the beam material viscosity in the contact zone without discontinuity of the target medium within this zone.

## Declarations

### Authors' contributions

Professors YAR and MVS formulated the problem, suggested the methods of solution and wrote the text of the manuscript. PhD student MGEM contributed in making mathematical treatment. All authors read and approved the final manuscript.

### Acknowledgements

The research described in this publication has been supported by the international project from the Russian Foundation for Basic Research No.14-08-92008-HHC-a and Taiwan National Science Council No. NSC 103-2923-E-011-002-MY3.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Landau LD, Lifshitz EM (1965) Theory of elasticity. In: A course of theoretical physics, vol 7. Nauka, Moscow (Engl. transl. by Pergamon Press in 1970)Google Scholar
- Popov II, Rossikhin YuA, Shitikova MV, Chang T-P (2015) Impact response of a viscoelastic beam considering the changes of its microstructure in the contact domain. Mech time-depend mater 19:455–481. doi:10.1007/s11043-015-9273-9 View ArticleGoogle Scholar
- Rabotnov YuN (2014) Equilibrium of an elastic medium with after-effect. Fract Calc Appl Anal 17(3):684–696. doi:10.2478/s13540-014-0193-1 View ArticleGoogle Scholar
- Rabotnov YuN (1966) Creep of structural elements. Nauka, Moscow (Engl. transl. by North-Holland, Amsterdam in 1969)Google Scholar
- Rossikhin YuA, Shitikova MV (1995) The ray method for solving boundary problems of wave dynamics for bodies having curvilinear anisotropy. Acta Mech 109:49–64View ArticleGoogle Scholar
- Rossikhin YuA, Shitikova MV (1996) The impact of elastic bodies upon beams and plates with consideration for the transverse deformations and extension of a middle surface. ZAMM 76(Suppl. 5):433–434Google Scholar
- Rossikhin YuA, Shitikova MV (1999) The impact of a sphere on a Timoshenko thin-walled beam of open section with due account for middle surface extension. ASME J Pressure Vessel Tech 121:375–383View ArticleGoogle Scholar
- Rossikhin YuA, Shitikova MV (2007a) Transient response of thin bodies subjected to impact: Wave approach. Shock Vibr Digest 39(4):273–309View ArticleGoogle Scholar
- Rossikhin YuA, Shitikova MV (2007b) The method of ray expansions for investigating transient wave processes in thin elastic plates and shells. Acta Mech 189:87–121View ArticleGoogle Scholar
- Rossikhin YuA, Shitikova MV (2013) Two approaches for studying the impact response of viscoelastic engineering systems: an overview. Comp Math Appl 66:755–773View ArticleGoogle Scholar
- Rossikhin YuA, Shitikova MV (2015) Fractional calculus: applications. Mathematics research developments series. In: Zeid Daou RA, Moreau X (eds) Features of fractional operators involving fractional derivatives and their applications to the problems of mechanics of solids, Chapter 8. Nova Science Publishers, New York, pp 165–226Google Scholar
- Timoshenko SP (1934) Theory of elasticity. McGraw-Hill, New YorkGoogle Scholar
- Vershinin VV (2014) Verification of analytical solution for a problem of high-velociy transversal impact on a prismatic beam. Appl Mech Mater 467:343–348. doi:10.4028/www.scientific.net/AMM.467.343 View ArticleGoogle Scholar