Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Ray Expansions in Dynamic Contact Problems

  • Yury A. Rossikhin†
  • Marina V. ShitikovaEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_98-1
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Synonyms

Definitions

The contact dynamic problems are those connected with nonstationary interactions of such thin bodies as plates and shells with a surrounding medium.

Preliminary Remarks

The problems connected with nonstationary interactions of thin shells with a surrounding medium belong to the contact dynamic problems and are of practical significance. These problems are the topical problems just as from the view point of fundamental investigations dealing with the contact dynamic problems, so also in the view of their applications to various fields of modern production, such as in problems in mining art when investigating the interaction of a working with a support, in petroleum extractive and refining industry in storage of oil and petroleum products in spherical subsurface vessels, in seismology and seismic survey when cavities of the Earth’s crust interacting with incident and reflected waves may play the role of resonators increasing the energy of wave fields in the vicinity of these cavities, and so on.

Mathematical methods used for solving contact dynamic problem are of a great variety; among them are the method of invariant functional solutions suggested by V.N. Smirnov and S.L. Sobolev, the Wiener-Hopf method, the integral transform method, the generalized Volterra and Hadamard methods, the method of characteristics, various numerical methods, and the ray method. The advantage of the last named method over other mentioned methods is in its pictorial compatibility with the intuitive notion of wave propagation, in the ease of interpretation of the results obtained, as well as in its versatility, since the ray series, which are the basis of this method, may be successfully applied both for solving the problems connected with the generation of harmonic vibrations and waves and for tackling the problems dealing with the origination of transient vibrations and waves.

Rossikhin (1978) was the first to use the ray method for solving contact dynamic problems when studying transient vibration of a plate resting on an elastic anisotropic half-space. A comprehensive review of papers, wherein dynamic contact problems were solved by the ray method based on the theory of discontinuities, can be found in Rossikhin and Shitikova (1995a). Further this method was extended for the solution of the problems dealing with nonlinear thin bodies. Thus, the transient vibrations of a nonlinear elastic plate resting on an elastic isotropic half-space was studied in Rossikhin and Shitikova (1996), but the transient vibrations of a nonlinearly elastic infinite cylindrical shell of constant thickness and spherical shell being in external contact with an elastic isotropic surrounding medium were investigated in Rossikhin and Shitikova (1999a, 2000), respectively.

In this entry, dynamic contact interaction of an infinite cylindrical shell with an unbounded elastic medium will be considered as the typical example of the utilization of the ray expansion method for solving dynamic contact problems, resulting in the formulation of the criterion of stability of a cylindrical shell compressed along its axis with respect to nonstationary dynamic excitations.

Fig. 1

Scheme of a cylindrical shell being in dynamic contact with an external elastic medium

Dynamic Contact Interaction of an Infinite Cylindrical Shell with an Unbounded Elastic Medium

Let us consider nonstationary vibrations of an infinitely long elastic circular cylindrical shell of the thickness h and radius a being in external welded or smooth contact with an unbounded elastic isotropic medium and experiencing uniform axial compression of the force Nz = const (Rossikhin and Shitikova, 1999b). The axial z-coordinate and angle φ-coordinate are chosen as the coordinates on the medial surface of the cylindrical shell (Fig. 1). The equations of vibrations of such a shell have the form Timoshenko (1936)
$$\displaystyle \begin{aligned} &\quad \,\frac{\partial^2U_z}{\partial z^2}{+}\frac{1-\nu_1}{2a^2}\, \frac{\partial^2U_z}{\partial\varphi^2}{+}\frac{1+\nu_1}{2a}\,\frac{\partial^2 U_\varphi}{\partial z\partial\varphi}{+}\frac{\nu_1}{a}\,\frac{\partial U_r}{\partial z}\\&{=}\,\gamma^2 \frac{\partial^2 U_z}{\partial t^2}-\frac{\gamma^2}{\rho_1 h} q_z, \end{aligned} $$
(1)
$$\displaystyle \begin{aligned} &\quad \,\frac{1}{a^2}\frac{\partial^2 U_\varphi}{\partial\varphi^2}+\frac{1-\nu_1}{2}\, \frac{\partial^2 U_\varphi}{\partial z^2}+\frac{1+\nu_1}{2a}\, \frac{\partial^2 U_z}{\partial z\partial\varphi}\\&\quad + \frac{1}{a^2}\,\frac{\partial U_r}{\partial\varphi}+ \alpha\left[2(1-\nu_1)\frac{\partial^2 U_\varphi}{\partial z^2}\right. +\left .\frac{1}{a^2}\frac{\partial^2 U_\varphi}{\partial\varphi^2}\right.\\&\qquad \qquad \qquad \quad \,\left.-(2-\nu_1) \frac{\partial^3 U_r}{\partial z^2\partial\varphi}-\frac{1}{a^2}\, \frac{\partial^3 U_r}{\partial\varphi^3}\right]\\ &= \gamma^2\frac{\partial^2 U_\varphi}{\partial t^2}- \frac{\gamma^2}{\rho_1h}q_\varphi, \end{aligned} $$
(2)
$$\displaystyle \begin{aligned} &\quad \,\frac{1}{a^2}\,\frac{\partial U_\varphi}{\partial\varphi}+ \frac{U_r}{a^2}+\frac{\nu_1}{a}\,\frac{\partial U_z}{\partial z}\\ &\quad + \alpha\left[a^2\varDelta^2 U_r-(2-\nu_1)\frac{\partial^3U_\varphi} {\partial z^2\partial\varphi}- \frac{1}{a^2}\,\frac{\partial^3 U_\varphi}{\partial\varphi^3}\right]\\ &\quad +\frac{\gamma^2}{\rho_1 h}N_z\frac{\partial^2 U_r}{\partial z^2}= -\gamma^2\frac{\partial^2 U_r}{\partial t^2}+\frac{\gamma^2}{\rho_1 h}q_r, \end{aligned} $$
(3)
where Ur, Uφ, and Uz are the shell median surface displacement components; qr, qφ, and qz are the external loading components along r, φ, and z-axes of the cylindrical system of coordinates, respectively; ρ1 is the material density; ν1 is the Poisson’s ratio; \(\gamma ^2=(1-\nu ^2_1)\rho _1E^{-1}_1\); E1 is the Young’s modulus; α = h2(12a2)−1; t is the time; and Δ = 2∂z2 + a−22∂φ2.
The dynamic behavior of the unbounded elastic isotropic medium surrounding the shell in the cylindrical system of r, φ, z coordinates is described by the following set of equations (Nowacki, 1970):
$$\displaystyle \begin{aligned} \frac{\partial\sigma_{rr}}{\partial r}+\frac{\sigma_{rr}-\sigma_{\varphi\varphi}}{r} + \frac{1}{r}\,\frac{\partial\sigma_{r\varphi}}{\partial\varphi} + \frac{\partial\sigma_{rz}}{\partial z} =\rho \frac{\partial^2u_r}{\partial t^2},\end{aligned} $$
(4)
$$\displaystyle \begin{aligned} \frac{1}{r}\,\frac{\partial\sigma_{\varphi\varphi}}{\partial\varphi}+ \frac{\partial\sigma_{r\varphi}}{\partial r}+ \frac{2\sigma_{r\varphi}}{r}+\frac{\partial\sigma_{\varphi z}}{\partial z}= \rho\frac{\partial^2u_\varphi}{\partial t^2},\end{aligned} $$
(5)
$$\displaystyle \begin{aligned} \frac{\partial\sigma_{zz}}{\partial z}+\frac{\partial\sigma_{rz}}{\partial r}+ \frac{\sigma_{rz}}{r}+\frac{1}{r}\,\frac{\partial\sigma_{\varphi z}}{\partial\varphi} = \rho\frac{\partial^2u_z}{\partial t^2},\end{aligned} $$
(6)
$$\displaystyle \begin{aligned} \sigma_{rr}=2\mu e_{rr}+\lambda e, \quad \sigma_{\varphi\varphi}= 2\mu e_{\varphi\varphi}+\lambda e,\end{aligned} $$
(7)
$$\displaystyle \begin{aligned} \quad \sigma_{zz}=2\mu e_{zz}+\lambda e, \end{aligned} $$
(8)
$$\displaystyle \begin{aligned} \sigma_{r\varphi}=2\mu e_{r\varphi}, \quad \sigma_{\varphi z}=2\mu e_{\varphi z}, \quad \sigma_{zr}=2\mu e_{zr}, \end{aligned} $$
(9)
$$\displaystyle \begin{aligned} e_{rr}=\frac{\partial u_r}{\partial r}, \quad e_{\varphi\varphi}=\frac{1}{r}\,\frac{\partial u_\varphi}{\partial\varphi}+ \frac{u_r}{r}, \quad e_{zz}=\frac{\partial u_z}{\partial z}, \end{aligned} $$
(10)
$$\displaystyle \begin{aligned} & e_{r\varphi}=\frac{1}{2}\left(\frac{1}{r}\,\frac{\partial u_r}{\partial\varphi} +\frac{\partial u_\varphi}{\partial r}-\frac{u_\varphi}{r}\right),\\ & e_{rz}=\frac{1}{2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right), \end{aligned} $$
(11)
$$\displaystyle \begin{aligned} e_{\varphi z}=\frac{1}{2}\left(\frac{\partial u_\varphi}{\partial z} + \frac{1}{r}\,\frac{\partial u_z}{\partial\varphi}\right), \quad e=e_{rr}+e_{\varphi\varphi}+e_{zz}, \end{aligned} $$
(12)
where σij and eij (i, j = r, φ, z) are the components of the stress and strain tensors, respectively, ui are the displacement vector components, ρ is the isotropic medium density, and λ and μ are Lamé constants.
The conditions of welded contact between the shell and the elastic medium at r = a
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \sigma_{rr}=q_r,\quad \sigma_{r\varphi}=q_\varphi,\quad \sigma_{rz}=q_z,\\ U_r=u_r,\quad U_\varphi=u_\varphi,\quad U_z=u_z, \end{array} \end{aligned} $$
(13)
or the conditions of smooth contact and the initial conditions
$$\displaystyle \begin{aligned} &\frac{\partial U_r}{\partial t}\bigg|{}_{t=0}=V_r^0(z,\varphi),\quad \frac{\partial U_\varphi}{\partial t}\bigg|{}_{t=0}=V_\varphi^0(z,\varphi),\\ &\frac{\partial U_z}{\partial t}\bigg|{}_{t=0}=V_z^0(z,\varphi), \end{aligned} $$
(15)
where \(V_r^0(z,\varphi )\), \(V_\varphi ^0(z,\varphi )\), and \(V_z^0(z,\varphi )\) are the given functions, should be added to Eqs. (1)–(12).

Solution for the Space with a Cylindrical Cavity

As a result of snap-action application of displacement velocities to the boundary of the cylindrical cavity, the longitudinal and transverse waves, on fronts of which the components of the stress and strain tensors experience a discontinuity, i.e., the wave fronts are the surfaces of strong discontinuity, propagate in the isotropic elastic medium. Behind the wave fronts, the solution for a certain function Z(r, φ, z, t) to be found is constructed in the form of a series in terms of powers \(y_{(\alpha )}=t-(r-a)G_{(\alpha )}^{-1}\ge 0\), i.e., the ray series (Rossikhin and Shitikova, 1995a, 2007)
$$\displaystyle \begin{aligned} &\quad Z(r,\varphi,z,t)\\&=\sum_{\alpha=1}^2\sum_{k=0}^\infty \frac{1}{k!}\left[Z_{,(k)}^{(\alpha)}\right]\Bigg|{}_{y_{(\alpha)}=0} y_{(\alpha)}^k H(y_{(\alpha)}), \end{aligned} $$
(16)
where \(\left [Z_{,(k)}^{(\alpha )}\right ]=\left (\partial ^kZ^{(\alpha )}/\partial t^k\right )^+ - \left (\partial ^kZ^{(\alpha )}/\partial t^k\right )^-\) are the jumps in the kth time derivatives of the function Z(α)(r, φ, z, t) on the fronts of the shock waves; the signs “+ ” and “−” denote that the given function is calculated immediately ahead of and behind the wave front, respectively; α signifies the ordinal number of the wave, α = 1 for the quasi-longitudinal wave and α = 2 for the quasi-transverse wave, respectively; G(α) are the shock wave velocities; and H(y(α)) is the unit Heaviside function.
To determine the coefficients of the ray series (16) for the desired functions σrr, σ, σrz, ur, uφ, and uz, Eqs. (4)–(6) should be differentiated k times and Eqs. (7)–(12) k + 1 times with respect to time. Then it is necessary to take their difference on the different sides of each of the wave surfaces, and to apply the condition of compatibility for discontinuities of the (k + 1)th derivatives, which, as it has been shown by Rossikhin and Shitikova (1995b), for the physical components of a certain vector or tensor takes the form (see details in the entry “Ray Expansion Theory”)
$$\displaystyle \begin{aligned} G\left[\partial Z_{,(k)}/\partial r\right]= -\left[Z_{,(k+1)}\right]+ \delta\left[ Z_{,(k)}\right]/\delta t, \end{aligned} $$
(17)
where δδt is the Thomas δ-derivative (Thomas, 1961).
As a result of straightforward calculations, the recurrent relationships are obtained in terms of the discontinuities in the partial time derivatives of the displacement velocities vr, vφ, and vz where an index without brackets after a point denotes the partial derivative with respect to the corresponding coordinate, but the functions Lk−1, Mk−1, and Nk−1 have the form
$$\displaystyle \begin{aligned} M_{k-1}&=\mu\frac{\delta}{\delta t}\left(r\frac{\delta[v_{\varphi,(k-1)}]}{\delta t}\right)\\ &\quad + G^2(\lambda+2\mu)\frac{1}{r}[v_{\varphi,(k-1)}]_{,\varphi\varphi}\\&\quad +G^2\mu r[v_{\varphi,(k-1)}]_{,zz}\\ &\quad +G^2(\lambda+\mu)[v_{z,(k-1)}]_{,\varphi z}\\&\quad +G(\lambda+\mu)\frac{\delta}{\delta t} [v_{r,(k-1)}]_{,\varphi}\\ &\quad +G^2(\lambda+3\mu)\frac{1}{r}[v_{r,(k-1)}]_{,\varphi}\\ &-G^2\mu\frac{1}{r}[v_{\varphi,(k-1)}],\end{aligned} $$
$$\displaystyle \begin{aligned} N_{k-1}&=\mu\frac{\delta}{\delta t}\left(r\frac{\delta [v_{z,(k-1)}]}{\delta t}\right)\\ &\quad +G^2(\lambda+2\mu)r[v_{z,(k-1)}]_{,zz}\\&\quad +G^2\mu\frac{1}{r}[v_{z,(k-1)}]_{,\varphi\varphi}\\ &\quad +G(\lambda+\mu)r\frac{\delta}{\delta t}\left([v_{r,(k-1)}]_{,z}\right)\\ &\quad + G^2(\lambda+\mu)[v_{\varphi,(k-1)}]_{,\varphi z}\\&\quad +G^2(\lambda+\mu)[v_{r,(k-1)}]_{,z}. \end{aligned} $$
Now putting k = −1, 0, 1, 2, … in Eqs. (18)–(20), the velocities of the transient waves and discontinuities in the desired values of the orders k ≥ 0 could be obtained. Thus, on the first wave \(\rho G^2_{(1)}=\lambda +2\mu \), r = G(1)t + a,
$$\displaystyle \begin{aligned} {[v_{\varphi,(0)}]}&=[v_{z,(0)}]=0, \\\quad {[v_{r,(0)}]}&=f_{(0)}(\varphi, z)r^{-1/2}, \\ {[v_{\varphi,(1)}]}&=-G_{(1)}f_{(0),\varphi}r^{-3/2},\\\quad {[v_{z,(1)}]}&=-G_{(1)}f_{(0),z}r^{-1/2},\\ {[v_{r,(1)}]}&=f_{(1)}(\varphi, z)r^{-1/2} \\ &\quad +\frac{1}{2}G_{(1)} \Bigg\{\left(\frac 34 f_{(0)} -f_{(0),\varphi\varphi}\right) r^{-3/2} \\ &\qquad \qquad \qquad +f_{(0),zz}r^{1/2}\Bigg\},... \end{aligned} $$
(21)
and on the second wave \(\rho G_{(2)}^2=\mu \), r  = G(2)t + a,
$$\displaystyle \begin{aligned} {[v_{r,(0)}]}&=0, \quad [v_{\varphi,(0)}]=g_{(0)}(\varphi, z)r^{-1/2},\\\quad {[v_{z,(0)}]}&=l_{(0)}(\varphi, z)r^{-1/2}, \\ {[v_{r,(1)}]}&=G_{(2)}g_{(0),\varphi}r^{-3/2}+G_{(2)}l_{(0),z}r^{-1/2},\\ {[v_{\varphi,(1)}]}&=g_{(1)}(\varphi, z)r^{-1/2} \\ &\quad +\frac{1}{2}G_{(2)} \Bigg\{ \left(\frac{3}{4}g_{(0)}-g_{(0),\varphi\varphi}\right)r^{-3/2} \\ &\qquad \qquad \qquad +g_{(0),zz}r^{1/2}\Bigg\},\\ {[v_{z,(1)}]}&=l_{(1)}(\varphi, z)r^{-1/2} \\ &\quad -\frac{1}{2}G_{(2)}\Bigg\{ \left(\frac{1}{4}l_{(0)} +l_{(0),\varphi\varphi}\right)r^{-3/2}\\ &\qquad \qquad \qquad -l_{(0),zz}r^{1/2}\Bigg\},... \end{aligned} $$
(22)
where f(n)(φ, z), g(n)(φ, z), and l(n)(φ, z) (n = 0, 1, 2, … ) are arbitrary functions which are determined from the initial conditions and conditions of contact, and only the discontinuities of the zero and first order are presented (the higher-order discontinuities could be found in Rossikhin and Shitikova 1999b).
Considering formulas (21) and (22), one can obtain at each fixed instant of the time the desired values ui and σir (i = r, φ, z) behind the fronts of the shock waves up to the contact boundary as follows: where
$$\displaystyle \begin{aligned} G_{(\alpha)}[\sigma_{rr,(k)}]&=-(\lambda+2\mu)[v_{r,(k)}]\\ {}&\quad +(\lambda+2\mu) \frac{\delta[v_{r,(k-1)}]}{\delta t}\\ {}&\quad +\lambda G_{(\alpha)}\left(\frac{1}{r}[v_{\varphi,(k-1)}]_{,\varphi}\right.\\ {} &\qquad \qquad \qquad +[v_{z,(k-1)}]_{,z}\\ {} &\qquad \qquad \qquad + \left.\frac{1}{r}[v_{r,(k-1)}]\right),\\ {}G_{(\alpha)}[\sigma_{rz,(k)}]&=-\mu[v_{z,(k)}]+\mu\frac{\delta[v_{z,(k-1)}]}{\delta t}\\ {}&\quad + \mu G_{(\alpha)}[v_{r,(k-1)}]_{,z},\\ {}G_{(\alpha)}[\sigma_{r\varphi,(k)}]&=-\mu[v_{\varphi,(k)}]+\mu\frac{\delta[v_{\varphi,(k-1)}]} {\delta t}\\ {}&\quad +\mu G_{(\alpha)}\frac{1}{r}[v_{r,(k-1)}]_{,\varphi}\\ {} &\quad -\mu G_{(\alpha)}\frac{1}{r}[v_{\varphi,(k-1)}]. \end{aligned} $$

Solution for a Circular Cylindrical Shell in the Case of Welded Contact with a Surrounding Medium

Writing relationships (23) and (24) on the contact boundary, i.e., at r = a, with account for Eqs. (13) results in
Differentiating Eq. (25) with respect to time, putting t equal to zero, and then considering the initial conditions (15) yield
$$\displaystyle \begin{aligned} f_{(0)}(\varphi,z)&=a^{1/2}V_r^0(\varphi,z),\\\quad g_{(0)}(\varphi,z)&=a^{1/2}V_{\varphi}^0(\varphi,z),\\\quad l_{(0)}(\varphi,z)&=a^{1/2}V_z^0(\varphi,z). \end{aligned} $$
(27)

Substituting (25) and (26) into Eqs. (1)–(3), equating terms at equal powers of t, and considering (27), all other unknown functions f(i), g(i) and l(i) (i = 1, 2, 3, …) required to construct the solution with a desired accuracy of ti could be determined.

Substituting then the expressions for the found functions f(i), g(i), and l(i) (i = 0, 1, 2, 3, …) into formulas (25), and supposing that in the relationships (15) and thus in (27)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle V_r^0(\varphi,z)=V_r\sin l_1 z\cos l_2\varphi, \\ &\displaystyle &\displaystyle V_\varphi^0(\varphi,z)=V_\varphi\sin l_1z\sin l_2\varphi, \\ &\displaystyle &\displaystyle V_z^0(\varphi,z)=V_z\cos l_1z\cos l_2\varphi, \end{array} \end{aligned} $$
(28)
where V r, V φ, V z, l1 and l2 are the given constant values, the relationships for Uk (k = r, φ, z) within an accuracy of t4 take the form (Rossikhin and Shitikova, 1999b)
$$\displaystyle \begin{aligned} &\\ {}U_r(\varphi,z,t)&=\Bigg\{V_rt -\frac{\rho G_{(1)}}{\rho_1h}V_r\frac{t^2}{2} +\Bigg[\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2V_r +\frac{l_1^2}{\rho_1h}N_zV_r +\frac{\lambda-2\mu}{2\rho_1ha}V_r\\ {}&\qquad +\frac{\rho G_{(2)}}{\rho_1h}(G_{(1)}-2G_{(2)})\left(l_2a^{-1}V_\varphi -l_1V_z\right)\\ {}&\qquad -\gamma^{-2}(l_2a^{-2}V_\varphi+a^{-2}V_r-l_1\nu_1 a^{-1}V_z)\\ {}&\qquad -\alpha \gamma^{-2}a^{-2}\left.\left((l_1^2a^2+l_2^2)^2V_r +(2-\nu_1)a^2l_1^2l_2V_\varphi +l_2^3V_\varphi \right)\right]\frac{t^3}{6} \\ {}&\qquad +\Bigg[-\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^3V_r -2\frac{\rho G_{(1)}l_1^2}{(\rho_1h)^2}N_zV_r -\frac{(\lambda-2\mu)\rho G_{(1)}}{(\rho_1h)^2a}V_r \\ {}&\qquad +\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2\left( G_{(1)} -G_{(1)}^2G_{(2)}^{-1}+2G_{(2)}\right)\left(l_2a^{-1}V_\varphi-l_1V_z\right)\\ {}&\qquad +\frac{\mu G_{(1)}}{2\rho_1ha}\left(\frac{G_{(1)}}{G_{(2)}}-1\right) \left(l_2a^{-1}V_\varphi+l_1V_z\right)\\ {}&\qquad +\frac{(\lambda+2\mu)G_{(2)}}{\rho_1h} \left(\frac{G_{(1)}}{G_{(2)}}-1\right)\left(l_2^2a^{-2}+l_1^2\right)V_r\\ {}&\qquad +\frac{\rho G_{(2)}}{\rho_1h}\gamma^{-2} \left(\frac{G_{(1)}}{G_{(2)}}+1\right)\left( l_2a^{-2}V_\varphi -\nu_1 a^{-1}l_1V_z +\alpha\left[(2-\nu_1)l_1^2 +a^{-2}l_2\right]V_\varphi \right)\\ {}&\qquad + \left.\left. 2\frac{\rho G_{(1)}}{\rho_1h}\gamma^{-2}a^{-2} \left(1 +\alpha (l_1^2a^2+l_2^2)^2\right)V_r\right]\frac{t^4}{24}\right\} \sin l_1z\cos l_2\varphi, \end{aligned} $$
(29)
$$\displaystyle \begin{aligned}U_\varphi(\varphi,z,t)&=\Bigg\{V_\varphi t -\frac{\rho G_{(2)}}{\rho_1h}V_\varphi\frac{t^2}{2} +\Bigg[\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2V_\varphi +\frac{\rho G_{(2)}}{\rho_1ha}(G_{(1)}-2G_{(2)})l_2V_r\\ {}&\qquad -\frac{3\mu}{2\rho_1ha}V_\varphi -\gamma^{-2}\left(\frac{l_2^2}{a^2}V_\varphi+\frac{1-\nu_1}{2}l_1^2V_\varphi -\frac{1+\nu_1}{2a}l_1l_2V_z+\frac{l_2}{a^2}V_r\right)\\ {}&\qquad -\left.\alpha \gamma^{-2}\left(2(1-\nu_1)l_1^2V_\varphi+l_2^2a^{-2}V_\varphi +(2-\nu_1)l_1^2l_2V_r+l_2^3a^{-2}V_r\right)\right]\frac{t^3}{6}\\ {}&\qquad +\Bigg[-\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^3V_\varphi +3G_{(2)}\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2a^{-1}V_\varphi\\ {}&\qquad +\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2\frac{l_2}{a}\left( G_{(1)} -G_{(1)}^2G_{(2)}^{-1}+2G_{(2)}\right)V_r\\ {}&\qquad -\frac{\mu G_{(1)}l_2}{2\rho_1ha^2}\left(\frac{G_{(1)}}{G_{(2)}}-1\right) V_r -\frac{\mu G_{(2)}l_2}{\rho_1ha} \left(\frac{G_{(1)}}{G_{(2)}}-1\right)\left(l_2a^{-1}V_\varphi -l_1V_z\right)\\ {}&\qquad +2\frac{\rho G_{(2)}}{\rho_1h}\gamma^{-2}\Bigg\{\frac{l_2^2}{a^2}V_\varphi +\frac{1-\nu_1}{2}l_1^2V_\varphi -\frac{1+\nu_1}{2a}l_1l_2V_z\\ {}&\qquad + \alpha \left[ 2(1-\nu_1)l_1^2+a^{-2}l_2^2\right]V_\varphi \Bigg\} +\frac{\rho G_{(2)}}{\rho_1h}\gamma^{-2}\left(\frac{G_{(1)}}{G_{(2)}}+1\right)\\ {}&\qquad \times \left(\frac{l_2}{a^2}(1+l_2^2\alpha) +l_1^2l_2\alpha(2-\nu_1)\right)V_r\Bigg]\frac{t^4}{24}\Bigg\} \sin l_1z\sin l_2\varphi, \end{aligned} $$
(30)
$$\displaystyle \begin{aligned} &\\ {}U_z(\varphi,z,t)&=\Bigg\{V_zt -\frac{\rho G_{(2)}}{\rho_1h}V_z\frac{t^2}{2} +\Bigg[\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2V_z -\frac{\rho G_{(2)}}{\rho_1h}(G_{(1)}-2G_{(2)})l_1V_r\\ &\qquad -\left.\frac{\mu}{2\rho_1ha}V_z-\gamma^{-2}\left(l_1^2V_z +\frac{1-\nu_1}{2a^2}l_2^2V_z -\frac{1+\nu_1}{2a}l_1l_2V_\varphi+\frac{\nu_1}{a}l_1V_r\right)\right] \frac{t^3}{6}\\ &\qquad +\Bigg[-\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^3V_z +G_{(2)}\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2a^{-1}V_z\\ &\qquad -\left(\frac{\rho G_{(2)}}{\rho_1h}\right)^2\left( G_{(1)} -G_{(1)}^2G_{(2)}^{-1}+2G_{(2)}\right)l_1V_r +\frac{\mu G_{(1)}}{2\rho_1ha}\left(\frac{G_{(1)}}{G_{(2)}}-1\right)l_1V_r\\ &\qquad +\frac{\mu G_{(2)}}{\rho_1h} \left(\frac{G_{(1)}}{G_{(2)}}-1\right)l_1\left(l_2a^{-1}V_\varphi -l_1V_z\right)\\ &\qquad +2\frac{\rho G_{(2)}}{\rho_1h}\gamma^{-2} \left(l_1^2V_z+\frac{1-\nu_1}{2a^2}l_2^2V_\varphi -\frac{1+\nu_1}{2a}l_1l_2V_\varphi \right)\\ &\qquad -\left.\left. \gamma^{-2}\frac{\nu_1}{a}\frac{\rho G_{(2)}}{\rho_1h} \left(\frac{G_{(1)}}{G_{(2)}}+1\right)l_1V_r\right]\frac{t^4}{24}\right\} \cos l_1z\cos l_2\varphi. \end{aligned} $$
(31)
For the sake of simplicity, assume that V φ = V z = 0, but V r≠0. In this case the relationships (29)–(31) are reduced to
$$\displaystyle \begin{aligned}U_r(\varphi,z,t) &= V_r\Bigg\{t -\frac{\rho G_{(1)}}{\rho_1h}\frac{t^2}{2}\\ &\qquad +\frac{l_1^2}{\rho_1h}\left(N_z-N_z^*\right)\frac{t^3}{6} +\frac{\rho G_{(1)}}{\rho_1h}\Bigg[-2\frac{l_1^2}{\rho_1h}(N_z-N_z^*)\\ &\quad + \left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2+G_{(1)}\left(G_{(1)}-G_{(2)}\right) \left(l_1^2+l_2^2a^{-2}\right)\Bigg]\frac{t^4}{24}\Bigg\} \sin l_1z\cos l_2\varphi,{} \end{aligned} $$
(32)
$$\displaystyle \begin{aligned}U_\varphi(\varphi,z,t)&=V_rl_2\left\{\left[\frac{\rho G_{(2)}}{\rho_1ha}(G_{(1)}-2G_{(2)}) -\gamma^{-2}\left(a^{-2}(1+l_2^2\alpha) +l_1^2\alpha(2-\nu_1)\right)\right]\frac{t^3}{6}\right.\\ &\quad +\frac{\rho G_{(2)}}{\rho_1h}\left[\frac{\rho G_{(2)}}{\rho_1ha} \left(G_{(1)}-G_{(1)}^2G_{(2)}^{-1}+2G_{(2)}\right) -\frac{G_{(1)}}{2a^2}(G_{(1)}-G_{(2)})\right.\\ &\quad + \left.\left.\gamma^{-2}\left(\frac{G_{(1)}}{G_{(2)}}+1\right)\left(a^{-2} (1+l_2^2\alpha)+l_1^2\alpha(2-\nu_1)\right)\right]\frac{t^4}{24}\right\}\sin l_1z\sin l_2\varphi, \qquad {} \end{aligned} $$
(33)
$$\displaystyle \begin{aligned}U_z(\varphi,z,t)&=V_rl_1\left\{-\left[\frac{\rho G_{(2)}}{\rho_1h}(G_{(1)}-2G_{(2)}) +\nu_1a^{-1}\gamma^{-2}\right]\frac{t^3}{6}\right.\\ &\quad +\frac{\rho G_{(2)}}{\rho_1h}\left[-\frac{\rho G_{(2)}}{\rho_1h} \left(G_{(1)}-G_{(1)}^2G_{(2)}^{-1}+2G_{(2)}\right) +\frac{G_{(1)}}{2a}(G_{(1)}-G_{(2)})\right.\\ &\quad -\left.\left.\frac{\nu_1}{a\gamma^2}\left(\frac{G_{(1)}}{G_{(2)}}+1\right) \right]\frac{t^4}{24}\right\}\cos l_1z\cos l_2\varphi,{} \end{aligned} $$
(34)
where
$$\displaystyle \begin{gathered}{} N_z^*=\frac{1}{l_1^2}\left[D{(l_1^2+l_2^2a^{-2})}^2+ \frac{\rho_1h}{\gamma^2a^2}-\kappa\right], \end{gathered} $$
(35)
$$\displaystyle \begin{gathered} D=\frac{E_1h^3}{12(1-\nu_1^2)}=\frac{\alpha \rho_1ha^2}{\gamma^2} \end{gathered} $$
is the cylindrical rigidity of the shell and
$$\displaystyle \begin{gathered} \kappa=\frac{\lambda-2\mu}{2a}+\frac{\rho^2G_{(1)}^2}{\rho_1h} \end{gathered} $$
is the coefficient revealing the influence of the surrounding medium.
If supposed that
$$\displaystyle \begin{aligned} N_z-N_z^*=\varDelta N_z, \end{aligned} $$
(36)
where ΔNz is a small positive value, then it follows from (32) that both coefficients standing in this formula in square brackets at t3 and t4 become positive values, i.e., the increase of the displacement component Ur is made possible as time goes on, which characterizes the unstable dynamic behavior of the cylindrical shell with respect to nonstationary loading.

Thus, the axial compression force \(N_z=N_z^*\) defined by the formula (35) is the critical force under nonstationary dynamic deformation of the shell being in welded external contact with an elastic isotropic medium.

Neglecting the influence of the surrounding medium in Eq. (35), i.e., putting κ = 0, yields
$$\displaystyle \begin{aligned} N_z^*=\frac{1}{l_1^2}\left[D{(l_1^2+l_2^2a^{-2})}^2+ \frac{\rho_1h}{\gamma^2a^2}\right]. \end{aligned} $$
(37)

From the comparison of the formulas (35) and (37), it follows that the axial critical force for the cylindrical shell which is in welded contact with the surrounding medium is less than the critical force for the free cylindrical shell. The reason is that the non-free shell is subjected to the transverse force of inertia from the contacting medium, which decreases the magnitude of the longitudinal critical force.

At l2 = 0, i.e., in the case of the symmetrical type of buckling of the shell, the critical force \(N_z^*\) (35) has the minimum magnitude
$$\displaystyle \begin{aligned} (N_z^*)_{\mathrm{min}}=2\sqrt{\left(\frac{\rho_1h}{\gamma^2a^2} -\kappa\right)D}. \end{aligned} $$
(38)
when
$$\displaystyle \begin{aligned} l_1^2=\sqrt{\left(\frac{\rho_1h}{\gamma^2a^2}-\kappa\right)\frac{1}{D}}. \end{aligned} $$
(39)
If no account has been taken of a surrounding medium, then the expression for Ur with the allowance made for the five terms of the ray series takes the form
$$\displaystyle \begin{aligned} U_r(\varphi,z,t) &= V_r\Bigg\{t+\frac{l_1^2}{\rho_1h}\left(N_z-N_z^*\right)\frac{t^3}{6} +\Bigg[\Bigg(\frac{l_1^2}{\rho_1h}\Bigg)^2\left(N_z-N_z^*\right)^2 +\left(\frac{\nu_1l_1}{\gamma^2a}\right)^2\\ &\quad +\left(\frac{l_2}{\gamma^2a^2}\right)^2\left(1+\alpha l_2^2+\alpha (2-\nu_1)l_1^2a^2\right)^2\Bigg] \frac{t^5}{120}\Bigg\}\sin l_1z \cos l_2\varphi,{} \end{aligned} $$
(40)
where the critical force is obtained from (37), and the minimal critical force for a free cylindrical shell at l2 = 0 has the form
$$\displaystyle \begin{aligned} (N_z^*)_{\mathrm{min}}=\frac{E_1h^2}{a(1-\nu_1^2)\sqrt{3}}, \qquad l_1^2=\frac{2\sqrt{3}}{ha}. \end{aligned} $$
(41)

The magnitude \((N_z^*)_{\mathrm {min}}\) defined by formula (41) is approximately equal to the critical force in the case of symmetrical buckling of the circular cylindrical shell under static loading (Timoshenko, 1936).

Neglecting the influence of the surrounding medium in Eq. (35) and considering that thin cylindrical shells under axial compression buckle along short longitudinal waves, i.e., l1 is large, then at l2≠0 for this more general case, once again formula (41) could be obtained for the critical force.

Thus, the method of power series with the variable t allows one to investigate dynamic stability of both free and non-free cylindrical shells with respect to nonstationary loading.

Solution for a Circular Cylindrical Shell in the Case of Smooth Contact with a Surrounding Medium

In the case of smooth contact (14), formula (25) is valid only at i = r. Using Eq. (25) differentiated with respect to time at i = r and Eq. (26) at i = φ, z with consideration for qφ = qz = 0 and putting t = 0 in the resulted relations with due account for the first initial condition from (15) yield
To make use of the rest of the initial conditions (15), the functions Uφ(φ, z, t) and Uz(φ, z, t) should be represented as
$$\displaystyle \begin{aligned} U_i(\varphi,z,t)=\sum_{k=1}^4\frac{1}{k!}U_{i(k)}t^k\quad (i=\varphi,z), \end{aligned} $$
(43)
where \(U_{i(1)}=V_i^0(\varphi ,z)\), but Ui(k) at k > 1 are as yet unknown functions in φ and z.

Substituting (25) at i = r, (43) at i = φ, z, and (26) into Eqs. (1)–(3), equating terms at equal powers of t, and regarding for (42), all other unknown functions f(1), f(2), g(1), l(1), Uφ(k), and Uz(k) (k = 2, 3, ...) required for the construction of the solution with an accuracy of tk could be determined (Rossikhin and Shitikova, 1999b).

Substituting then found expressions f(j), g(j), l(j) (j = 0, 1, 2, ..) into (25) at i = r and (43) at i = φ, z and considering that the functions \(V_i^0(\varphi ,z)\) (i = r, φ, z) are given in the form (28) yield the following relationships for Uk (k = r, φ, z):
$$\displaystyle \begin{aligned} U_r(\varphi,z,t)&=\Bigg\{V_rt -\frac{\rho G_{(1)}}{\rho_1h}V_r\frac{t^2}{2} +\Bigg[\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2V_r +\frac{l_1^2N_z}{\rho_1h}V_r\\ &\quad +\frac{\lambda-2\mu}{2\rho_1ha}V_r -\gamma^{-2}(l_2a^{-2}V_{\varphi}+a^{-2}V_r -l_1\nu_1 a^{-1}V_z)\\ &\quad -\alpha \gamma^{-2}a^{-2}\left((l_1^2a^2+l_2^2)^2V_r +(2-\nu_1)l_1^2l_2V^0_\varphi -l_2^3V_\varphi\Bigg)\right]\frac{t^3}{6} \\ &\quad +\Bigg[-\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^3V_r -2\frac{\rho G_{(1)}l_1^2}{(\rho_1h)^2}N_zV_r -\frac{(\lambda-2\mu)\rho G_{(1)}}{(\rho_1h)^2a}V_r\\ &\quad +\frac{2\rho G_{(1)}G_{(2)}}{\rho_1h} \left(G_{(1)}-G_{(2)}\right)\left(l_2^2a^{-2}+l_1^2\right)V_r\\ &\quad +\frac{\lambda}{\rho_1h} \left(G_{(1)}-2G_{(2)}\right)\left(l_2^2a^{-2}+l_1^2\right)V_r\\ &\quad +\frac{\rho G_{(1)}}{\rho_1h}\gamma^{-2} \left(l_2a^{-2}V_\varphi-\nu_1 a^{-1}l_1V_z +\alpha(2-\nu_1)l_1^2l_2V_\varphi +\alpha a^{-2}l_2^3V_\varphi\right)\\ &\quad +\left.2\frac{\rho G_{(1)}}{\rho_1h}\gamma^{-2}a^{-2} \left(1+\alpha(l_1^2a^2+l_2^2)^2\right)V_r \right]\frac{t^4}{24}\Bigg\} \sin l_1z\cos l_2\varphi, \quad \end{aligned} $$
(44)
$$\displaystyle \begin{aligned} U_\varphi(\varphi,z,t)&=\left\{V_\varphi t +\left[-\gamma^{-2}\left(a^{-2}l_2^2V_\varphi +\frac{1-\nu_1}{2}l_1^2V_\varphi-\frac{1+\nu_1}{2a}l_1l_2V_z+l_2a^{-2}V_r\right)\right.\right. \\ &\quad -\left.\alpha \gamma^{-2}\left(2(1-\nu_1)l_1^2V_\varphi +a^{-2}l_2^2V_\varphi +(2-\nu_1)l_1^2l_2V_r+a^{-2}l_2^3V_r\right)\right]\frac{t^3}{6}\\ &\quad + \left.\frac{\rho G_{(1)}}{\rho_1h}\gamma^{-2}a^{-2}l_2\left[1 +\alpha \left(2(1-\nu_1)l_1^2a^2+l_2^2\right)\right]V_r \frac{t^4}{24}\right\}\sin l_1z\sin l_2\varphi,\qquad\end{aligned} $$
(45)
$$\displaystyle \begin{aligned} U_z(\varphi,z,t)&=\left\{V_zt+\left[-\gamma^{-2}\left(l_1^2V_z +\frac{1-\nu_1}{2a^2}l_2^2V_z-\frac{1+\nu_1}{2a}l_1l_2V_\varphi -\frac{\nu_1}{a}l_1V_r\right)\right]\frac{t^3}{6}\right. \\ &\quad - \left.\frac{\nu_1}{a}\frac{\rho G_{(1)}}{\rho_1h}\gamma^{-2}l_1V_r \frac{t^4}{24}\right\}\cos l_1z\cos l_2\varphi. \end{aligned} $$
(46)
For the sake of simplicity, assume that V φ = V z = 0, but V r≠0. In this case the relationships (44)–(46) are reduced to
$$\displaystyle \begin{aligned} U_r(\varphi,z,t)&=V_r\Bigg\{t -\frac{\rho G_{(1)}}{\rho_1h}\frac{t^2}{2} +\frac{l_1^2}{\rho_1h}\left(N_z-N_z^*\right)\frac{t^3}{6} +\frac{\rho G_{(1)}}{\rho_1h}\Bigg[-\frac{2l_1^2}{\rho_1h}(N_z-N_z^*) +\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2\\ &\quad + \left.\left. (l_1^2+l_2^2a^{-2})\left(2G_{(2)}(G_{(1)}-G_{(2)}) +\frac{\lambda(G_{(1)}-2G_{(2)})}{\rho G_{(1)}} \right)\right]\frac{t^4}{24}\right\} \sin l_1z\cos l_2\varphi,\qquad\end{aligned} $$
(47)
$$\displaystyle \begin{gathered}{}U_\varphi=-V_r\frac{l_2}{\gamma^2a^2}\left[1+\alpha\left(l_2^2 +(2-\nu_1)l_1^2a^2\right)\right]\left(\frac{t^3}{6} -\frac{\rho G_{(1)}}{\rho_1h}\frac{t^4}{24}\right)\sin l_1z\sin l_2\varphi, \end{gathered} $$
(48)
$$\displaystyle \begin{aligned}U_z=V_r\frac{\nu_1l_1}{a\gamma^2}\left(\frac{t^3}{6} -\frac{\rho G_{(1)}}{\rho_1h}\frac{t^4}{24}\right) \cos l_1z\cos l_2\varphi, \end{aligned} $$
(49)
where \(N_z^*\) is defined by (35).
Since the relationship
$$\displaystyle \begin{aligned} 2\rho G_{(1)}G_{(2)}(G_{(1)}-G_{(2)})+\lambda(G_{(1)}-2G_{(2)})\;>0, \end{aligned}$$
then in deciding on Nz in the form (36), it is seen from Eq. (47) that both coefficients standing in this formula at t3 and t4 become positive values, i.e., the increase of the displacement component Ur is made possible as time goes on, which characterizes the unstable dynamic behavior of the cylindrical shell with respect to nonstationary loading.

Thus, the axial compression force \(N_z=N_z^*\) defined by formula (35) is the critical force under nonstationary dynamic deformation of the shell being in smooth external contact with an elastic isotropic medium.

Solutions of Simplified Equations for Circular Cylindrical Shells

Now let us consider the simplified version of the linear theory concerning the cases when buckling of shells is accompanied by the appearance of rather fine waves whose dimensions, if only in one direction, are small compared with the shell dimensions. As this takes place, the shell within each bulge can be considered as the shallow shell. For this purpose the following differential equations could be utilized (Donnell, 1976):
$$\displaystyle \begin{aligned} D\varDelta^2w+\rho_1h\frac{\partial^2w}{\partial t^2}+N_z\frac{\partial^2w}{\partial z^2} =\frac{h}{a}\frac{\partial^2\chi}{\partial z^2}+q, \end{aligned} $$
(50)
$$\displaystyle \begin{aligned} \frac{1}{E_1}\varDelta^2\chi=-\frac{1}{a}\frac{\partial^2w}{\partial z^2},\end{aligned} $$
(51)
where w = Ur is the normal displacement of the shell middle surface, χ is the stress function, q = qr is the intensity of the transverse load, and qφ = qz = 0.
Eliminating the value χ from Eqs. (50) and (51) results in the differential equation for solving the problem on stability
$$\displaystyle \begin{aligned} D\varDelta^4w&+\frac{E_1h}{a^2}\,\frac{\partial^4w}{\partial z^4} +\rho_1h\varDelta^2\left(\frac{\partial^2w}{\partial t^2}\right)\\ &+N_z\varDelta^2\left(\frac{\partial^2w}{\partial z^2}\right) =\varDelta^2q. \end{aligned} $$
(52)
subjected to the initial condition for the deflection velocity in the form
$$\displaystyle \begin{aligned} \frac{\partial w}{\partial t}\Big|{}_{t=0}=V\sin l_1z\cos l_2\varphi, \end{aligned} $$
(53)
where V = V r.
Utilizing the ray expansions within an accuracy of t4, the solution for w(φ, z, t) has the form (Rossikhin and Shitikova, 1999b)
$$\displaystyle \begin{aligned} w(\varphi,z,t)&=V\left\{t-\frac{\rho G_{(1)}}{\rho_1h}\frac{t^2}{2} +\frac{l_1^2}{\rho_1h}\left(N_z-N_z^*\right)\frac{t^3}{6} +\frac{\rho G_{(1)}}{\rho_1h}\left[-2\frac{l_1^2}{\rho_1h}(N_z-N_z^*)\right.\right.\\ &\quad +\left.\left. \left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2 +(l_1^2+l_2^2a^{-2}) \left(2G_{(2)}(G_{(1)}-G_{(2)})+\frac{\lambda(G_{(1)}-2G_{(2)})}{\rho G_{(1)}} \right)\right]\frac{t^4}{24}\right\}\\ &\quad \times \sin l_1z\cos l_2\varphi, \end{aligned} $$
(54)
where
$$\displaystyle \begin{aligned} N_z^*&=\frac{1}{l_1^2}\left[D\left(l_1^2+\frac{1}{a^2}l_2^2\right)^2\right.\\ &\qquad +\left.\frac{E_1hl_1^4}{a^2}\left(l_1^2+\frac{1}{a^2}l_2^2\right)^{-2} -\kappa\right]. \end{aligned} $$
(55)
Putting κ = 0 in (55) provides the formula
$$\displaystyle \begin{aligned} N_z^*&=D\left(l_1^2+\frac{1}{a^2}l_2^2\right)^2l_1^{-2}\\ &\quad +\frac{E_1h}{a^2}\left(l_1^2+\frac{1}{a^2}l_2^2\right)^{-2} l_1^2, \end{aligned} $$
(56)
from which the known formula for the critical force under static loading of a cylindrical shell could be obtained (Timoshenko, 1936)
$$\displaystyle \begin{aligned} (N_z^*)_{\mathrm{min}}=\frac{E_1h^2}{a\sqrt{3(1-\nu_1^2)}}. \end{aligned} $$
(57)
There exists another simplified version of the linear theory concerned with the case of undulation weakly expressed along the length of the shell, which can be used for investigating shells of moderate and especially considerable length. This version assumes the middle surface to be unstretched in the hoop direction and neglects shears in the middle surface. Transverse forces and bending moments in the axial direction, as well as twisting moments, are considered to be equal to zero. In this case, the following differential equation for solving the problems on stability is obtained (Handbook, 1968):
$$\displaystyle \begin{aligned} D\varOmega\varOmega w+E_1h\frac{\partial^4w}{\partial z^4} =N_z\left(\frac{\partial^4w}{\partial\varphi^2\partial z^2} -\frac{\partial^6w}{\partial\varphi^4\partial z^2}\right)\frac{1}{a^2} -\frac{\rho_1h}{a^2}\,\frac{\partial^6 w}{\partial\varphi^4\partial t^2} +\frac{1}{a^2}\,\frac{\partial^4q}{\partial\varphi^4}, \end{aligned} $$
(58)
subjected to the initial condition (53), where
$$\displaystyle \begin{aligned} \varOmega=\left(\frac{\partial^4}{\partial\varphi^4}+\frac{\partial^2}{\partial\varphi^2}\right)\frac{1}{a^3}. \end{aligned}$$
The four-term ray expansion solution for w(φ, z, t) takes the form (Rossikhin and Shitikova, 1999b)
$$\displaystyle \begin{aligned} w(\varphi,z,t)&=V\left\{t-\frac{\rho G_{(1)}}{\rho_1h}\frac{t^2}{2} +\frac{l_1^2(1+l_2^2)}{\rho_1hl_2^2}\left(N_z-N_z^*\right)\frac{t^3}{6} +\frac{\rho G_{(1)}}{\rho_1h}\Bigg[-2\frac{l_1^2(1+l_2^2)}{\rho_1hl_2^2}(N_z-N_z^*)\right.\\ &\quad +\left(\frac{\rho G_{(1)}}{\rho_1h}\right)^2 +(l_1^2+l_2^2a^{-2})\left(2G_{(2)}(G_{(1)}-G_{(2)})\right.\\ &\quad +\left.\left.\left.\frac{\lambda(G_{(1)}-2G_{(2)})}{\rho G_{(1)}} \right)\right]\frac{t^4}{24}\right\}\sin l_1z\cos l_2\varphi, \qquad\end{aligned} $$
(59)
where
$$\displaystyle \begin{aligned} N_z^*=\frac{l_2^2}{l_1^2(l_2^2+1)}\left[\frac{D(l_2^2-1)^2}{a^4} +\frac{E_1ha^2l_1^4}{l_2^4}-\kappa\right]. \end{aligned} $$
(60)
From the condition of minimization for the critical force \(N_z^*\) with respect to \(l=a^2l_1^2\), it follows
$$\displaystyle \begin{aligned} l^2=\frac{l_2^2}{\sqrt{E_1h}} \left[\frac{E_1h^3}{12a^2(1-\nu_1^2)}{(l_2^2-1)}^2 -\kappa a^2\right]^{1/2}, \end{aligned} $$
(61)
as well as
$$\displaystyle \begin{aligned} (N_z^*)_{\mathrm{min}}=\frac{2\sqrt{E_1h}}{l_2^2+1}\left[\frac{E_1h^3{(l_2^2-1)}^2}{12a^2(1-\nu_1^2)} -\kappa a^2\right]^{1/2}. \end{aligned} $$
(62)
Putting κ = 0 in (61) and (62) yields the known relationships (Timoshenko, 1936)
$$\displaystyle \begin{aligned} l^2&=\frac{h}{2a\sqrt{3}}\,\frac{l_2^2(l_2^2-1)}{\sqrt{1-\nu_1^2}},\\\qquad(N_z^*)_{\mathrm{min}}&=\frac{E_1h^2}{a\sqrt{3(1-\nu_1^2)}}\,\frac{l_2^2-1}{l_2^2+1}. \end{aligned} $$
(63)

Numerical Investigations

In order to investigate the time t dependence of the value Ur = w for three types of cylindrical shells described by formulas (32), (47), (54), and (59) at various magnitudes of the axial compressive load Nz, it is convenient to rewrite these relations into the dimensionless form with due account for five terms of the ray series. As a result, the following relationships could be obtained:

in the case of welded contact (32)
$$\displaystyle \begin{aligned} \widetilde U_r&=\left\{\tilde t-r_1 \frac{\tilde t^2}{2} + \varDelta \widetilde N_z\frac{\tilde t^3}{6} +r_1\left[-2\varDelta \widetilde N_z+r_1^2+(1-g)h_1^2d_1\right]\frac{\tilde t^4}{24} +\left[\varDelta \widetilde N_z^2+r_1\varDelta \widetilde N_z-r_1^4\right.\right.\\ &\quad +(\nu_1dr_1lh_1^2)^2+(l_2dr_1h_1^2)^2(1+\alpha d_2)^2-\frac 12 r_1^2h_1^2(1-4g^2) +2r_1^2h_1^3d\left(1+\alpha(l_2^4-l^4)\right) \\ &\quad -\frac 12 r_1h_1^3(l_2^2-l^2)g(1-g)-r_1^2(1-g)(2+g-2g^2)h_1^2d_1 -\frac 12 r_1h_1^3(1-g)d_1\\ &\quad +\left.\left. r_1^2h_1^2g(1-2g)(1-2g^2)d_1\right]\frac{\tilde t^5}{120} \right\}\sin l_1z \cos l_2\varphi,\qquad\end{aligned} $$
(64)
in the case of smooth contact (47)
$$\displaystyle \begin{aligned} \widetilde U_r&=\left\{\tilde t- r_1 \frac{\tilde t^2}{2}+ \varDelta \widetilde N_z\frac{\tilde t^3}{6} +r_1\left[-2\varDelta \widetilde N_z+r_1^2+h_1^2d_1(1-4g^2+4g^3)\right]\frac{\tilde t^4}{24}\right. \\ &\quad +\left[\varDelta \widetilde N_z^2+r_1^2\varDelta \widetilde N_z-r_1^4-\frac 12 r_1^2h_1^2 (1-4g^2)-2r_1^2h_1^3(1-\alpha d_1)d\right.\\ &\quad -r_1^2h_1^3l^2\nu_1(1+\nu_1)(1-2g^2)-r_1^2h_1^2d_1(1+2g-6g^2+4g^3)+(r_1\nu_1dlh_1^2)^2\\ &\quad +\left.\left. (l_2h_1^2r_1d)^2(1+\alpha d_2)^2 -r_1^2l_2^2h_1^3d(1+\alpha d_2)(1-2g^2)\right]\frac{\tilde t^5}{120}\right\} \sin l_1z \cos l_2\varphi, \qquad\end{aligned} $$
(65)
in the case of the first variant of the simplified shell equation (54)
$$\displaystyle \begin{aligned} \widetilde U_r&=\left\{\tilde t- r_1 \frac{\tilde t^2}{2} + \varDelta \widetilde N_z\frac{\tilde t^3}{6} +r_1\left[-2\varDelta \widetilde N_z+r_1^2+h_1^2d_1(1-4g^2+4g^3)\right]\frac{\tilde t^4}{24}\right.\\ &\quad +\left[\varDelta \widetilde N_z^2+r_1^2\varDelta \widetilde N_z-r_1^4 +4r_1^2h_1^3dd_1l_2^2\alpha -\frac 12 r_1l^2h_1^3(1-2g)^2\right.\\ &\quad +2r_1^2h_1^3d(1-\nu_1^2)(l^2-l_2^2)l^4d_1^{-3} -\frac 12 r_1^2h_1^2(1-4g^2)+2r_2^2h_1^2d_1g(1-2g^2) \\ &\quad -\left.\left.\frac 12 r_1l_2^2h_1^3(1-2g^2)(1+4g-6g^2)+2r_1^2h_1^2d_1(1-g^2)\right] \frac{\tilde t^5}{120}\right\}\sin l_1z \cos l_2\varphi, \quad \end{aligned} $$
(66)
in the case of the second variant of the simplified free shell equation (59)
$$\displaystyle \begin{aligned} \widetilde w&=\left\{\tilde t- r_1 \frac{\tilde t^2}{2} + \varDelta \varDelta\widetilde N_z\frac{1+l_2^2}{l_2^2}\frac{\tilde t^3}{6} +r_1\left[-2\varDelta \widetilde N_z\frac{1+l_2^2}{l_2^2} +r_1^2+d_1h_1^2(1-4g^2+4g^3)\right]\frac{\tilde t^4}{24}\right.\\ &\quad +\left[\left(\frac{1+l_2^2}{l_2^2}\right)^2\varDelta \widetilde N_z^2+\frac{1+l_2^2}{l_2^2}r_1^2\varDelta \widetilde N_z -r_1^4-2r_1^2d_1h_1^2(1-g-g^2+2g^3)\right. \\ &\quad -\frac 12 l_2^2h_1^3r_1(1-2g^2)(1+4g-6g^2)-\frac 12 r_1l^2h_1^3(1-2g^2)^2 +4\alpha h_1^3r_1^2d(l_2^2-1)\\ &\quad - \left.\left. 2d(1-\nu_1^2)r_1^2h_1^3l^4l_2^{-4}-\frac 12 h_1^2r_1^2(1-4g^2)\right] \frac{\tilde t^5}{120}\right\}\sin l_1z \cos l_2\varphi, \qquad\end{aligned} $$
(67)
in the case of the free classical shell (40)
$$\displaystyle \begin{gathered}{} \widetilde w=\left\{\tilde t+\varDelta \widetilde N_z\frac{\tilde t^3}{6} +\left[\varDelta \widetilde N_z^2+\left(\frac{\nu_1}{1-\nu_1^2}lh^2_1\right)^2 +\left(\frac{l_2}{1-\nu_1^2}h^2_1(1+\alpha d_2)\right)^2 \right]\frac{\tilde t^5}{120}\right\}\sin l_1z\cos l_2\varphi, \end{gathered} $$
(68)
in the case of the first variant of the simplified free shell equation
$$\displaystyle \begin{aligned} \widetilde w=\left\{\tilde t+\varDelta \widetilde N_z\frac{\tilde t^3}{6} +\varDelta \widetilde N_z^2 \frac{\tilde t^5}{120}\right\}\sin l_1z\cos l_2\varphi,\end{aligned} $$
(69)
and in the case of the second variant of the simplified free shell equation
$$\displaystyle \begin{aligned} \widetilde w&=\left\{\tilde t+\frac{1+l_2^2}{l_2^2}\varDelta \widetilde N_z\frac{\tilde t^3}{6}\right.\\ &\qquad \left.+\left(\frac{1+l_2^2}{l_2^2}\right)^2 \varDelta \widetilde N_z^2 \frac{\tilde t^5}{120}\right\}\sin l_1z\cos l_2\varphi,\end{aligned} $$
(70)
where r1 = ρρ1, g = G(2)∕G(1), l = l1a, h1 = ha, \(d_1=l_2^2+l^2\), \(d_2=l_2^2+(2+\nu _1)l^2\), and \(d=\frac {1}{1-\nu _1^2}\frac {E_1}{\lambda +2\mu }\).
In formulas (64)–(67) and in formulas (68)–(70), the dimensional and corresponding dimensionless values are connected, respectively, by the following relations: where G = (E1ρ1)1∕2.

Figures 2, 3 and 4 present the curves corresponding to the dimensionless time dependence of the dimensionless radial displacement of the shell for various magnitudes of the value \(\varDelta \widetilde N_z\) calculated due to Eqs. (64)–(70) for the following magnitudes of the parameters: r1 = 0.25, g = 0.4, h1 = 0.1, E1∕(λ + 2μ) = 8.25, ν1 = 0.3, l2 = 44, \(l_2^2=10\), and \(\sin l_1z=\sin l_2\varphi =\sqrt {2}/2\).

Fig. 2

The dimensionless radial displacement of the shell as function of the dimensionless time: (a) welded contact (64) and (b) smooth contact (65). Digits near curves denote the magnitudes of the value \(\varDelta \widetilde N_z\)

Fig. 3

The dimensionless radial displacement of the shell as function of the dimensionless time: (a) first variant of the simplified shell equation (66) and (b) second variant of the simplified shell equation (67). Digits near curves denote the magnitudes of the value \(\varDelta \widetilde N_z\)

Fig. 4

The dimensionless radial displacement of the free shell as function of the dimensionless time: (a) classical shell (68), (b) first variant of the simplified shell equation (69), and (c) second variant of the simplified shell equation (70). Digits near curves denote the magnitudes of the value \(\varDelta \widetilde N_z\)

Reference to Figs. 2, 3 and 4 shows that at various magnitudes of \(\varDelta \widetilde N_z\), the curves descriptive of the time dependence of the small deflection behave differently. Some of them describe oscillating process; the others represent aperiodic process. What actually happens is that all of them describe oscillating motion, with the only difference that for some magnitudes of \(\varDelta \widetilde N_z\), it is sufficient to limit by five terms of the power series with respect to time to observe this, but for the other magnitudes of \(\varDelta \widetilde N_z\), five-term truncated series are inadequate to detect this fact, and we are thus led to take more terms of the series into account. However, no matter how many terms of the power series one is led to take into consideration, the general tendency in the behavior of shell deflections with time is evident: a maximum of the shell deflection on the time interval equal to a half of the period of oscillations increases with increase in the magnitude of the value \(\varDelta \widetilde N_z\) holding small nonstationary loading unchanged. It is clear that for some magnitudes of the value \(\varDelta \widetilde N_z\), the shell deflection at some of its points may exceed the limiting magnitudes resulting in crack formation and fracture of the shell.

Conclusion

Formulation of the problem on dynamic stability of a cylindrical shell in contact with an external elastic medium with respect to external nonstationary excitations differs fundamentally from classical statement of the similar problem on dynamic stability, since the latter deals only with external harmonic excitations and therefore fails to account for the initial conditions.

Thus, the approach described in this entry allows investigations to be made on the dynamic stability and instability of a cylindrical shell under axial compression with respect to nonstationary loading on its internal surface is suggested. It is based on the behavior of the ray expansion of the compressed shell radial displacement at the first instants of the time after the onset of nonstationary excitation. This method enables one to determine the critical load of axial compression dependent on the initial conditions.

Based on the analysis of the calculating data, the criterion of stability of a cylindrical shell compressed along its axis with respect to nonstationary dynamic excitations can be formulated. This criterion is founded on violating the alternation of signs of the power series describing the time dependence of the shell deflection when the magnitudes of the compression force are greater than a certain critical value. In so doing the vibrational regime of the shell is not changed by the aperiodic motion, but merely the increase of maximum values of the shell’s dynamic deflection is observed at discrete points during the time interval equal to a half of the period of oscillations from the onset of the motion. If this maximal value of the dynamic deflection is greater than the limiting values of the shell deflection, which are determined from the strength analysis as an example, then one can consider that under the given compressed force, the shell loses its stability, i.e., the criterion of stability is not absolute one, as it takes place in the problems of static or dynamic stability with respect to harmonic excitations, but it is the relative criterion.

Thus, the ray method put forward enables one to solve, with considerable success, dynamic problems on the snap-action interaction of both linearly elastic and nonlinearly elastic thin-walled bodies with a surrounding medium. The analytical character of the solution obtained both for the shell and the medium is one of the merits of this method.

Cross-References

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Center for Wave Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia

Section editors and affiliations

  • Marina V. Shitikova
    • 1
  1. 1.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia