Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Boundary-Value Dynamic Problems of Thin Bodies, Ray Expansion Approach

  • Yury A. Rossikhin
  • Marina V. ShitikovaEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_108-1



The nonstationary boundary-value problems of thin bodies are those connected with the transient wave propagation in thin bodies in the form of wave surfaces of strong or weak discontinuities generated by the external excitation.

Preliminary Remarks

It is known that the models of thin bodies based on the Kirchhoff-Love hypothesis allow one to achieve reasonable accuracy when solving a set of practical problems. However, in some cases this scheme is found to be incomplete. This is particularly true for the dynamic processes in thin bodies connected with the propagation of strain waves. The case in point is the deformation excited by impact or otherwise way in a certain zone of the thin body and subsequently imparted to different directions along its middle surface by means of wave motions. In this case, being within the limits of the classical Kirchhoff-Love concepts and having regard to the forces of inertia corresponding to the displacements tangent to the middle surface, one would describe the wave processes connected with shortening-elongation of the thin body element in the middle surface; but in so doing, the description of the transmission of transverse forces and transverse deformations associated with the local action of the suddenly applied normal loads is kept away from the consideration. As this takes place, two factors are going in importance: the shear deformations connected with the transverse forces and the rotary inertia of the thin body’s elements. Taking into account these factors in addition to the “classical” deformations and forces of inertia leads to the fact that in this case the equations of motion of thin bodies become of the hyperbolic type. A related model of thin bodies is usually associated with the name of Stephen Timoshenko who suggested it as applied to the theory of bending of beams (Timoshenko, 1928). Further works in the field applied to plates and shells are due to Uflyand (1948), Mindlin (1951, 1961), Reissner (1945), Naghdi (1957), Ambartsumjan (1961), and others.

One of the limitations of the Timoshenko theory is that in governing equations, there exists a certain correction factor k (shear coefficient) which is determined experimentally, and depending on the character of the experiment, it can take on different magnitudes. Thus, Reissner (1945) and Ambartsumjan (1961) put k2 = 5∕6; the values k2 = 2∕3 and 8∕9 can be found in Timoshenko (1928). The first of these magnitudes is also given by Uflyand (1948). Mindlin (1951) has suggested to determine k2 for plates reasoning from comparison of the elastic wave velocity on the basis of the accepted model with the corresponding velocity found by virtue of the three-dimensional (3D) equations of the theory of elasticity. Then the magnitude of k2 ranges from 0.76 to 0.91 with variation of the Poisson’s ratio σ between 0 and 0.5. Sometimes the magnitude k2 = π2∕12 is in use, which is obtained from comparison of the frequencies of the first antisymmetric mode of vibrations of a rectangular extended plate found by the strict theory and by the relationships taking shear and rotary inertia into account and so on.

An alternative method of constructing the basic relationships of the theory of thin bodies is to expand the displacements or stresses into the series (power or functional) with respect to the normal coordinate and to hold a certain truncated series depending on the required accuracy and the character of a problem (Murty, 1970; Reddy, 1984). Substituting these series into the boundary conditions on internal and external surfaces of a thin body results in differential equations, but substitution into the 3D equations of elasticity leads to recurrent symbolic relationships allowing one to determine all coefficients of the higher-order expansions. Under this approach, particular values entering by artificial means (as, for instance, the shear factor in the Timoshenko model and its generalizations) are absent in the coefficients of equations. However, the cumbersome mathematical treatment and the severity of equations and boundary conditions are the essential drawbacks of this approach.

The ray method suggested in Rossikhin and Shitikova (2007) for solving transient dynamic problems resulting in the propagation of surfaces of strong discontinuity in thin bodies and presented in this entry is also free from the values entering by artificial means, since it is based on the reduction of the 3D equations of the dynamic theory of elasticity written in discontinuities to the two-dimensional (2D) equations by virtue of integration over the coordinate perpendicular to the middle surface of a thin body. The simplicity and clearness inherent in the “classical” theory of thin bodies remain in the process.

Problem Formulation and Governing Equations

Suppose that a surface-strip of strong discontinuity propagates in a thin body, which is generated as a result of transient excitation of the surface of the thin body or its boundary. The 3D equations of the dynamic theory of elasticity in the Cartesian rectangular set of the x1−, x2 −, and x3 −coordinates have the form
$$\displaystyle \begin{gathered}{} \dot\sigma_{ij}=\lambda v_{l,l}\delta_{ij}+\mu(v_{i,j}+v_{j,i}), \end{gathered} $$
$$\displaystyle \begin{gathered} {} \sigma_{ij,j}=\rho \dot v_i, \end{gathered} $$
where σij and vi are the stress tensor and velocity vector components, respectively, λ and μ are Lamé coefficients; ρ is the density; an overdot denotes a partial time derivative; a Latin index after a point labels the partial derivative with respect to the corresponding spatial coordinates x1, x2, and x3; Latin indices take on the magnitudes of 1, 2, and 3; and the summation is carried out over the repeating indices.
Differentiating the relationships (1) and (2) k times with respect to time t, writing the obtained equations on the different sides of the wave surface, and taking their difference yield
$$\displaystyle \begin{gathered}{} [\sigma_{ij, (k+1)}]\,{=}\,\lambda [v_{l,l(k)}]\delta_{ij}\,{+}\,\mu([v_{i,j(k)}]\,{+}\,[v_{j,i(k)}]), \end{gathered} $$
$$\displaystyle \begin{gathered} {} [\sigma_{ij,j(k)}]=\rho [v_{i,(k+1)}], \end{gathered} $$
where [Z,(k)] = (kZ∂tk)+ − (kZ∂tk) and the signs “+ ” and “−” refer to the magnitudes of the derivative kZ∂tk of the certain desired function Z calculated before and behind the wave surface, respectively.

In further discussion instead of the wave surface-strip, it is convenient to consider the wave line L, which is the intersection of the wave surface-strip with the middle surface of the thin body; this line propagates along the solid body’s middle surface with the normal velocity G of the wave surface-strip. The normal trajectories of motion of the points of the wave line L (normal trajectories are the lines at each point of which the normals to the wave line coincide with the tangents to them) are the rays along which a disturbance propagates with the velocity G, i.e., they are the geodetic lines on the middle surface. Introduce the set of the s1, s2, ξ coordinates connected with the rays in the following manner: s1 is the arc length measured along the ray, s2 is the arc length measured along the wave line, and ξ is the distance along the normal to the middle surface of the body (Fig. 1).

In order to obtain the recurrent equations, the following conditions of compatibility (Thomas, 1961) should be used (see the entry “Ray Expansions Theory” for details):
$$\displaystyle \begin{aligned}{}[v_{i,j(k)}]&=-G^{-1}[v_{i,(k+1)}]\lambda_j +\frac{\delta[v_{i,(k)}]}{\delta s_1}\lambda_j\\ &\quad +\frac{\delta[v_{i,(k)}]}{\delta s_2}\tau_j +\frac{\delta[v_{i,(k)}]}{\delta \xi}\xi_j, \end{aligned} $$
$$\displaystyle \begin{aligned}{}[\sigma_{ij,j(k)}]&=-G^{-1}[\sigma_{ij,(k+1)}]\lambda_j +\frac{\delta[\sigma_{ij,(k)}]}{\delta s_1}\lambda_j\\ &\quad +\frac{\delta[\sigma_{ij,(k)}]}{\delta s_2}\tau_j +\frac{\delta[\sigma_{ij,(k)}]}{\delta \xi}\xi_j, \end{aligned} $$
where λi, ξi, and τi are the unit vector components: tangent to the ray λ, its main normal ξ (the main normal of the geodetic line ξ is orthogonal to the middle surface of the thin body at each point (McConnel, 1957)), and its binormal τ, respectively, δδs1 and δδs2 are the absolute derivatives on the middle surface, and δδξ is the absolute derivative with respect to the ξ −coordinate. Since x1, x2, and x3 are the Cartesian coordinates, then the absolute derivatives of the spatial tensors coincide with the ordinary derivatives.
Fig. 1

Scheme of the location of the wave and the ray on the shell’s middle surface with the corresponding basic vectors

Rewriting Eqs. (3) and (4) in the ray coordinates s1, s2, and ξ with due account for (5) and (6) yields
$$\displaystyle \begin{aligned}{}[\sigma_{ij, (k+1)}]&=-G^{-1}\lambda[v_{l,(k+1)}]\lambda_l\delta_{ij} -G^{-1}\mu([v_{i,(k+1)}]\lambda_j+[v_{j,(k+1)}]\lambda_i)\\ {}&\quad +\lambda\left(\frac{\delta[v_{l,(k)}]}{\delta s_1}\lambda_l +\frac{\delta[v_{l,(k)}]}{\delta s_2}\tau_l +\frac{\delta[v_{l,(k)}]}{\delta \xi}\xi_l\right)\delta_{ij}\\ &\quad +\mu\left(\frac{\delta[v_{i,(k)}]}{\delta s_1}\lambda_j + \frac{\delta[v_{j,(k)}]}{\delta s_1}\lambda_i +\frac{\delta[v_{i,(k)}]}{\delta s_2}\tau_j \right.\\ &\quad + \left.\frac{\delta[v_{j,(k)}]}{\delta s_2}\tau_i +\frac{\delta[v_{i,(k)}]}{\delta \xi}\xi_j +\frac{\delta[v_{j,(k)}]}{\delta \xi}\xi_i\right), \end{aligned} $$
$$\displaystyle \begin{aligned} &\quad -G^{-1}[\sigma_{ij, (k+1)}]\lambda_j +\frac{\delta[\sigma_{ij,(k)}]}{\delta s_1}\lambda_j\\ &\quad +\frac{\delta[\sigma_{ij,(k)}]}{\delta s_2}\tau_j +\frac{\delta[\sigma_{ij,(k)}]}{\delta \xi}\xi_j\\ &=\rho [v_{i,(k+1)}]. \end{aligned} $$
Hereafter the values [vi,(k)] entering in relationships (7) and (8) are expanded in terms of the ray coordinates λ, τ, and ξ
$$\displaystyle \begin{aligned}{}[v_{i,(k)}]=\omega_{(k)}\lambda_i+\theta_{(k)}\tau_i+\eta_{(k)}\xi_i, \end{aligned} $$
where ω(k) = [vi,(k)]λi, θ(k) = [vi,(k)]τi, and η(k) = [vi,(k)]ξi.
The condition
$$\displaystyle \begin{aligned}{}[\sigma_{ij, (k)}]\xi_i\xi_j=0, \quad -\;\frac{h}{2}<\xi<\frac{h}{2}, \end{aligned} $$
which is usually used in the theory of thin bodies, should be added to (7) and (8), where h is the thin body’s thickness.

The equality (10) implies that on the wave surface, the discontinuities in the normal stresses in the direction of the normal to the middle surface could be neglected as compared with the discontinuities in the main stresses.

The condition (10) leads to the following relationship:
$$\displaystyle \begin{aligned}{}[\epsilon_{\xi,(k+1)}]&=\frac{\lambda}{(\lambda+2\mu)G}\;[v_{l,(k+1)}]\lambda_l\\ &\quad -\frac{\lambda}{\lambda+2\mu}\;\frac{\delta[v_{l,(k)}]}{\delta s_1}\;\lambda_l\\ &\quad -\frac{\lambda}{\lambda+2\mu}\;\frac{\delta[v_{l,(k)}]}{\delta s_2}\;\tau_l, \end{aligned} $$
$$\displaystyle \begin{aligned} [\epsilon_{\xi,(k+1)}]=\frac{\delta([v_{l,(k)}]\xi_l)}{\delta \xi}.\end{aligned} $$

The relationship (11) should be considered as the formula for defining the unknown value [𝜖ξ,(k)], which together with the values ω(k), θ(k), and η(k) form a set of four unknown functions, as opposed to a three-dimensional medium, where there are only three unknown functions.

Comparing this assumption (10) with that of the Uflyand-Mindlin theory (Uflyand, 1948; Mindlin, 1951) about the smallness of the normal stress perpendicular to the plate’s middle surface, one can see the analogy between these two assumptions. However, in spite of the analogy between these assumptions, there exists a difference between these two hypotheses, namely, the plate thickness h remains constant in the Uflyand-Mindlin theory, but in our case, the plate thickness might change on the wave front by a small value (11).

The validity of the condition (10), as it will be shown later, is confirmed by the fact that it provides the wanted velocity of the longitudinal wave in a thin body, which differs from the longitudinal wave velocity in a 3D medium.

For further mathematical treatment, it is convenient to project Eqs. (7) and (8) on the ray coordinates λ, τ, and ξ with due account for (9) and Frenet formulae for rays and wave lines (Rossikhin and Shitikova, 2007); in so doing the value [𝜖ξ,(k)] is to be eliminated by virtue of formula (11):
$$\displaystyle \begin{aligned}{}[\sigma_{\lambda\lambda (k+1)}]&= [\sigma_{ij, (k+1)}]\lambda_i\lambda_j=-G^{-1}\rho G_1^2\;\omega_{(k+1)} +\rho G_1^2\left(\frac{\delta\omega_{(k)}}{\delta s_1} -\text{\ae}_1\eta_{(k)}\right)\\ {}&\quad +(\rho G_1^2-2\rho G_2^2)\left\{\frac{\delta\theta_{(k)}}{\delta s_2} -\text{\ae}_2(\eta_{(k)}\cos\theta+\omega_{(k)}\sin\theta)\right\}, \quad \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} [\sigma_{\tau\tau (k+1)}]&\displaystyle =&\displaystyle [\sigma_{ij, (k+1)}]\tau_i\tau_j=-G^{-1}(\rho G_1^2-2\rho G_2^2)\omega_{(k+1)} \\ {}&\displaystyle &\displaystyle + (\rho G_1^2-2\rho G_2^2)\left(\frac{\delta\omega_{(k)}}{\delta s_1} -\text{\ae}_1\eta_{(k)}\right)\\ {}&\displaystyle &\displaystyle +\rho G_1^2\left\{\frac{\delta\theta_{(k)}}{\delta s_2} -\text{\ae}_2(\eta_{(k)}\cos\theta+\omega_{(k)}\sin\theta)\right\},\quad \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} [\sigma_{\lambda\tau (k+1)}]&\displaystyle =&\displaystyle [\sigma_{ij, (k+1)}]\lambda_i\tau_j=-G^{-1}\mu\;\theta_{(k+1)} \\ &\displaystyle &\displaystyle +\mu\left(\frac{\delta\theta_{(k)}}{\delta s_1} +\frac{\delta\omega_{(k)}}{\delta s_2}+2\tau_1\eta_{(k)}+\text{\ae}_2\theta_{(k)}\sin\theta\right), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} [\sigma_{\lambda\xi (k+1)}]&\displaystyle =&\displaystyle [\sigma_{ij, (k+1)}]\lambda_i\xi_j=-G^{-1}\mu\;\eta_{(k+1)}\\ &\displaystyle &\displaystyle + \mu\left(\frac{\delta\eta_{(k)}}{\delta s_1}+\frac{\delta\omega_{(k)}}{\delta\xi} -\tau_1\theta_{(k)}+\text{\ae}_1\omega_{(k)}\right), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned}{}[\sigma_{\tau\xi (k+1)}]=[\sigma_{ij, (k+1)}]\tau_i\xi_j=\mu\left(\frac{\delta\eta_{(k)}}{\delta s_2} +\frac{\delta\theta_{(k)}}{\delta\xi} -\tau_1\omega_{(k)}+\text{\ae}_2\theta_{(k)}\cos\theta\right),\end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \rho\omega_{(k+1)}&\displaystyle =&\displaystyle \rho [v_{i,(k+1)}]\lambda_i =-G^{-1}[\sigma_{ij, (k+1)}]\lambda_i\lambda_j\\ &\displaystyle &\displaystyle +\frac{\delta}{\delta s_1} \left([\sigma_{ij, (k)}]\lambda_i\lambda_j\right) +\frac{\delta}{\delta s_2} \left([\sigma_{ij, (k)}]\lambda_i\tau_j\right)\\ &\displaystyle &\displaystyle -\text{\ae}_1\left([\sigma_{ij, (k)}]\xi_i\lambda_j +[\sigma_{ij, (k)}]\lambda_i\xi_j\right) -[\sigma_{ij, (k)}]\left(\tau_2^g\xi_i\tau_j-\text{\ae}_2\tau_i\tau_j\sin\theta\right) \\ &\displaystyle &\displaystyle -[\sigma_{ij, (k)}]\text{\ae}_2\left(\lambda_i\xi_j\cos\theta +\lambda_i\lambda_j\sin\theta\right) +\frac{\delta}{\delta \xi} \left([\sigma_{ij, (k)}]\lambda_i\xi_j\right), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \rho\theta_{(k+1)}&\displaystyle =&\displaystyle \rho [v_{i,(k+1)}]\tau_i =-G^{-1}[\sigma_{ij, (k+1)}]\tau_i\lambda_j\\ &\displaystyle &\displaystyle +\frac{\delta}{\delta s_1} \left([\sigma_{ij, (k)}]\tau_i\lambda_j\right) +\frac{\delta}{\delta s_2} \left([\sigma_{ij, (k)}]\tau_i\tau_j\right)\\ &\displaystyle &\displaystyle -[\sigma_{ij, (k)}]\left(\text{\ae}_1\xi_j\tau_i -\tau_1\lambda_j\xi_i\right)\\ &\displaystyle &\displaystyle -[\sigma_{ij, (k)}]\text{\ae}_2\left(\xi_i\tau_j\cos\theta +\lambda_i\tau_j\sin\theta +\tau_i\xi_j\cos\theta+\tau_i\lambda_j\sin\theta\right) \\ &\displaystyle &\displaystyle +\frac{\delta}{\delta \xi} \left([\sigma_{ij, (k)}]\tau_i\xi_j\right), \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \rho\eta_{(k+1)}&\displaystyle =&\displaystyle \rho [v_{i,(k+1)}]\xi_i =-G^{-1}[\sigma_{ij, (k+1)}]\xi_i\lambda_j\\ &\displaystyle &\displaystyle +\frac{\delta}{\delta s_1} \left([\sigma_{ij, (k)}]\xi_i\lambda_j\right) +\frac{\delta}{\delta s_2} \left([\sigma_{ij, (k)}]\xi_i\tau_j\right)\\ &\displaystyle &\displaystyle -[\sigma_{ij, (k)}]\left(\tau_1\tau_i\lambda_j -\text{\ae}_1\lambda_i\lambda_j\right)\\ &\displaystyle &\displaystyle -[\sigma_{ij, (k)}]\left(\text{\ae}_2\xi_i\lambda_j\sin\theta -\tau_2^g\lambda_i\tau_j-\text{\ae}_2\tau_i\tau_j\cos\theta\right), \end{array} \end{aligned} $$
where G1 and G2 are the velocities of the quasi-longitudinal-flexural and quasi-twisting-shear waves, respectively, defined as
$$\displaystyle \begin{aligned} \rho G_1^2=\frac{E}{1-\sigma^2},\qquad\rho G_2^2=\frac{E}{2(1+\sigma)}=\mu, \end{aligned} $$
E is the modulus of elasticity; æ1 and τ1 are the curvature and the torsion of the ray, respectively, æ2 and τ2 are the curvature and the torsion of the wave line, respectively, and \(\tau _2^g=\tau _2+\delta \theta /\delta s_2\) is the geodetic torsion.

Note that the velocity G1 defined by Eq. (20) coincides with that of the longitudinal wave in the Uflyand-Mindlin theory of elastic plates (Uflyand, 1948; Mindlin, 1951).

During the deduction of (12)–(19), the use has been made of the fact that the ray, which is the geodetic line on the middle surface of the thin body, is orthogonal to the wave line, and hence the sum of their geodetic torsions is equal to zero (McConnel, 1957), i.e.,
$$\displaystyle \begin{aligned} \tau_2^g=-\;\tau_1. \end{aligned} $$
The deduction of the recurrent equations of the ray theory developed is based on the following assumed velocity displacement field:
$$\displaystyle \begin{aligned} \omega_{(k)} &=\omega_{(k)}^0+\xi \omega_{(k)}^1, \quad \theta_{(k)} =\theta_{(k)}^0+\xi \theta_{(k)}^1,\\ \eta_{(k)} &=\eta_{(k)}^0+\xi \eta_{(k)}^1, \end{aligned} $$
where the values \(\omega _{(k)}^0\), \(\omega _{(k)}^1\), \(\theta _{(k)}^0\), \(\theta _{(k)}^1\), \(\eta _{(k)}^0\), and \(\eta _{(k)}^1\) are ξ-independent.
Integrating (12)–(19) over ξ from − h∕2 to h∕2, then integrating the condition (10) and Eqs. (12)–(19) multiplied by ξ also over ξ from − h∕2 to h∕2 with due account for expansions (22), and further eliminating the generalized forces
$$\displaystyle \begin{aligned} N_{\lambda(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij(k+1)}]\lambda_i\lambda_jd\xi,\\ N_{\tau(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij,(k+1)}]\tau_i\tau_jd\xi,\\ N_{\lambda\tau(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij,(k+1)}]\lambda_i\tau_jd\xi,\\ Q_{\lambda(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij, (k)}]\xi_i\lambda_jd\xi,\\ Q_{\tau(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij, (k)}]\tau_i\xi_jd\xi, \\ M_{\lambda(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij(k+1)}]\lambda_i\lambda_j\xi d\xi,\\ M_{\tau(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij,(k+1)}]\tau_i\tau_j\xi d\xi,\\ M_{\lambda\tau(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij,(k+1)}]\lambda_i\tau_j\xi d\xi,\\ Q_{\lambda\xi(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij, (k)}]\xi_i\lambda_j\xi d\xi,\\ Q_{\tau\xi(k+1)}&=\int_{-h/2}^{h/2}[\sigma_{ij, (k)}]\tau_i\xi_j\xi d\xi\end{aligned} $$
acting in the median surface of the thin body from the resulting relationships, the following recurrent equations could be obtained (Rossikhin and Shitikova, 2007):
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_1^2)\omega_{(k+1)}^0= -G^{-1}\rho G_1^2 \left(2\frac{\delta\omega_{(k)}^0}{\delta s_1}-\text{\ae}_2\sin\theta\;\omega_{(k)}^0\right) +F_{i(k-1)}^0\lambda_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_2^2)\theta_{(k+1)}^0= -G^{-1}\rho G_2^2\left(2 \frac{\delta\theta_{(k)}^0}{\delta s_1} -\text{\ae}_2\sin\theta\;\theta_{(k)}^0+3\tau_1\eta_{(k)}^0\right) +F_{i(k-1)}^0\tau_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_2^2)\eta_{(k+1)}^0 =-G^{-1}\rho G_2^2 \left(2\frac{\delta\eta_{(k)}^0}{\delta s_1} -\text{\ae}_2\sin\theta\;\eta_{(k)}^0-3\tau_1\theta_{(k)}^0\right) +F_{i(k-1)}^0\xi_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_1^2)\omega_{(k+1)}^1= -G^{-1}\rho G_1^2 \left(2\frac{\delta\omega_{(k)}^1}{\delta s_1}-\text{\ae}_2\sin\theta\;\omega_{(k)}^1\right) +F_{i(k-1)}^1\lambda_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_2^2)\theta_{(k+1)}^1= -G^{-1}\rho G_2^2\left(2 \frac{\delta\theta_{(k)}^1}{\delta s_1}-\text{\ae}_2\sin\theta\;\theta_{(k)}^1+3\tau_1\eta_{(k)}^1\right) +F_{i(k-1)}^1\tau_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} G^{-2}(\rho G^2-\rho G_2^2)\eta_{(k+1)}^1 =-G^{-1}\rho G_2^2 \left(2\frac{\delta\eta_{(k)}^1}{\delta s_1} -\text{\ae}_2\sin\theta\;\eta_{(k)}^1-3\tau_1\theta_{(k)}^1\right) +F_{i(k-1)}^1\xi_i, \end{gathered} $$

where the functions \(F_{i(k-1)}^0\lambda _i\), \(F_{i(k-1)}^0\tau _i\), \(F_{i(k-1)}^0\xi _i\), \(F_{i(k-1)}^1\lambda _i\), \(F_{i(k-1)}^1\tau _i\), and \(F_{i(k-1)}^1\xi _i\) depend on the (k − 1)-order discontinuities of the desired values (Rossikhin and Shitikova, 2007).

The set of six recurrent Eqs. (24)–(29) allows one to determine six unknown values: \(\omega _{(k)}^0\), \(\omega _{(k)}^1\), \(\theta _{(k)}^0\), \(\theta _{(k)}^1\), \(\eta _{(k)}^0\), and \(\eta _{(k)}^1\), which are the functions of s1 and s2.

Putting G = G1 defined by (20) in (24)–(29), for the quasi-longitudinal wave Σ1, it is obtained
$$\displaystyle \begin{gathered}{} \frac{\delta\omega_{(k)}^0}{\delta s_1}-\;\frac{1}{2}\text{\ae}_2\sin\theta\;\omega_{(k)}^0 =\frac{1}{2}(\rho G_1)^{-1}F_{i(k-1)}^0\lambda_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \frac{\delta\omega_{(k)}^1}{\delta s_1}-\;\frac{1}{2}\text{\ae}_2\sin\theta\;\omega_{(k)}^1 =\frac{1}{2}(\rho G_1)^{-1}F_{i(k-1)}^1\lambda_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \theta_{(k)}^0=G_2^2G_1(G_1^2-G_2^2)^{-1} \left(-2\frac{\delta\theta_{(k-1)}^0}{\delta s_1} +\text{\ae}_2\sin\theta\;\theta_{(k-1)}^0-3\tau_1\eta_{(k-1)}^0 + G_1(\rho G_2^2)^{-1}F_{i(k-2)}^0\tau_i\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \theta_{(k)}^1=G_2^2G_1(G_1^2-G_2^2)^{-1} \left(-2\frac{\delta\theta_{(k-1)}^1}{\delta s_1} +\text{\ae}_2\sin\theta\;\theta_{(k-1)}^1 -3\tau_1\eta_{(k-1)}^1 + G_1(\rho G_2^2)^{-1}F_{i(k-2)}^1\tau_i\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \eta_{(k)}^0=G_2^2G_1(G_1^2-G_2^2)^{-1} \left(-2\frac{\delta\eta_{(k-1)}^0}{\delta s_1} +\text{\ae}_2\sin\theta\;\eta_{(k-1)}^0 +3\tau_1\theta_{(k-1)}^0 +G_1(\rho G_2^2)^{-1}F_{i(k-2)}^0\xi_i\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \eta_{(k)}^1=G_2^2G_1(G_1^2-G_2^2)^{-1} \left(-2\frac{\delta\eta_{(k-1)}^1}{\delta s_1} +\text{\ae}_2\sin\theta\;\eta_{(k-1)}^1 +3\tau_1\theta_{(k-1)}^1 +G_1(\rho G_2^2)^{-1}F_{i(k-2)}^1\xi_i\right). \end{gathered} $$
Integrating (30) and (31) yields
$$\displaystyle \begin{gathered}{} \omega_{(k)}^0=\varrho f_{(k)}^0+\;\frac{1}{2}(\rho G_1)^{-1}\varrho \int_{s_1^0}^{s_1}\frac{1}{\varrho}F_{i(k-1)}^0\lambda_i ds_1,\quad \varrho=\exp\left(\frac{1}{2} \int_{s_1^0}^{s_1}\text{\ae}_2\sin\theta\; ds_1\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \omega_{(k)}^1=\varrho f_{(k)}^1+\;\frac{1}{2}(\rho G_1)^{-1}\varrho \int_{s_1^0}^{s_1}\frac{1}{\varrho}F_{i(k-1)}^1\lambda_i ds_1, \end{gathered} $$

where \(f_{(k)}^0\) and \(f_{(k)}^1\) are arbitrary functions dependent only on s2.

The recurrent relationships (36), (37), and (32)–(35) determine on the quasi-longitudinal-flexural wave the values \(\omega _{(k)}^0\), \(\omega _{(k)}^1\), \(\theta _{(k)}^0\), \(\theta _{(k)}^1\), \(\eta _{(k)}^0\), and \(\eta _{(k)}^1\), and hence all generalized forces (23) at k = 0, 1, 2, … within the accuracy of the arbitrary functions \(f_{(k)}^i=f_{(k)}^i(s_2)\) (i = 0, 1).

Putting G = G2 defined by (20) in (24)–(29), for the quasi-twisting-transverse wave Σ2, yields
$$\displaystyle \begin{gathered}{} \frac{\delta\theta_{(k)}^0}{\delta s_1}-\;\frac{1}{2}\text{\ae}_2\sin\theta\;\theta_{(k)}^0+\;\frac 32\tau_1\eta_{(k)}^0 =\frac{1}{2}(\rho G_2)^{-1}F_{i(k-1)}^0\tau_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \frac{\delta\eta_{(k)}^0}{\delta s_1} -\;\frac{1}{2}\text{\ae}_2\sin\theta\;\eta_{(k)}^0 -\;\frac 32\tau_1\theta_{(k)}^0 =\frac{1}{2}(\rho G_2)^{-1}F_{i(k-1)}^0\xi_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \frac{\delta\theta_{(k)}^1}{\delta s_1}-\;\frac{1}{2}\text{\ae}_2\sin\theta\;\theta_{(k)}^1+\;\frac 32\tau_1\eta_{(k)}^1 =\frac{1}{2}(\rho G_2)^{-1}F_{i(k-1)}^1\tau_i, \end{gathered} $$
$$\displaystyle \begin{gathered} {} \frac{\delta\eta_{(k)}^1}{\delta s_1} -\;\frac{1}{2}\text{\ae}_2\sin\theta\;\eta_{(k)}^1 -\;\frac 32\tau_1\theta_{(k)}^1 =\frac{1}{2}(\rho G_2)^{-1}F_{i(k-1)}^1\xi_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \omega_{(k)}^0=G_1^2G_2(G_2^2-G_1^2)^{-1} \left(-2\frac{\delta\omega_{(k-1)}^0}{\delta s_1} +\text{\ae}_2\sin\theta\;\omega_{(k-1)}^0 + G_2(\rho G_1^2)^{-1}F_{i(k-2)}^0\lambda_i\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \omega_{(k)}^1=G_1^2G_2(G_2^2-G_1^2)^{-1} \left(-2\frac{\delta\omega_{(k-1)}^1}{\delta s_1} +\text{\ae}_2\sin\theta\;\omega_{(k-1)}^1 + G_2(\rho G_1^2)^{-1}F_{i(k-2)}^1\lambda_i\right). \end{gathered} $$
Integrating the set of Eqs. (38)–(41) yields
$$\displaystyle \begin{aligned} \theta_{(k)}^0 &= \varrho(g_{(k)}^0\sin\psi+h_{(k)}^0\cos\psi)\\ &\quad +\frac{1}{2}(\rho G_2)^{-1}\varrho\left\{\sin\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^0\tau_i\sin\psi +F_{i(k-1)}^0\xi_i\cos\psi\right) ds_1\right.\\ &\quad +\left. \cos\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^0\tau_i\cos\psi -F_{i(k-1)}^0\xi_i\sin\psi\right) ds_1\right\}, \end{aligned} $$
$$\displaystyle \begin{aligned} \eta_{(k)}^0 &=\rho(g_{(k)}^0\cos\psi-h_{(k)}^0\sin\psi)\\ &\quad +\frac{1}{2}(\rho G_2)^{-1}\varrho\left\{\cos\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^0\tau_i\sin\psi +F_{i(k-1)}^0\xi_i\cos\psi\right) ds_1\right.\\ &\quad -\left. \sin\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^0\tau_i\cos\psi -F_{i(k-1)}^0\xi_i\sin\psi\right) ds_1\right\}, \end{aligned} $$
$$\displaystyle \begin{aligned} \theta_{(k)}^1 &= \varrho(g_{(k)}^1\sin\psi+h_{(k)}^1\cos\psi)\\ &\quad +\frac{1}{2}(\rho G_2)^{-1}\varrho\left\{\sin\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^1\tau_i\sin\psi +F_{i(k-1)}^1\xi_i\cos\psi\right) ds_1\right.\\ &\quad +\left. \cos\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^1\tau_i\cos\psi -F_{i(k-1)}^1\xi_i\sin\psi\right) ds_1\right\}, \end{aligned} $$
$$\displaystyle \begin{aligned} \eta_{(k)}^1 &=\varrho(g_{(k)}^1\cos\psi-h_{(k)}^1\sin\psi)\\ &\quad +\frac{1}{2}(\rho G_2)^{-1}\varrho\left\{\cos\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^1\tau_i\sin\psi +F_{i(k-1)}^1\xi_i\cos\psi\right) ds_1\right.\\ &\quad -\left. \sin\psi \int_{s_1^0}^{s_1}\frac{1}{\varrho}\left(F_{i(k-1)}^1\tau_i\cos\psi -F_{i(k-1)}^1\xi_i\sin\psi\right) ds_1\right\}, \end{aligned} $$
where \(g_{(k)}^i\) and \(h_{(k)}^i \quad (i=0,1)\) are arbitrary functions dependent only on s2 and
$$\displaystyle \begin{aligned} \psi= \frac 32 \int_{s_1^0}^{s_1} \tau_2^g ds_1. \end{aligned}$$

The recurrent relationships (44)–(47) and (42)–(43) define on the quasi-transverse wave the values \(\omega _{(k)}^0\), \(\omega _{(k)}^1\), \(\theta _{(k)}^0\), \(\theta _{(k)}^1\), \(\eta _{(k)}^0\), and \(\eta _{(k)}^1\), and hence all generalized forces (23) at k = 0, 1, 2, … within the accuracy of the arbitrary functions \(g_{(k)}^i=g_{(k)}^0(s_2)\) and \(h_{(k)}^i=h_{(k)}^0(s_2)\quad (i=0,1)\).

Note that at \(\tau _2^g=\tau _1=0\), the value ψ = 0, and the solution of (38) and (39) takes the form
$$\displaystyle \begin{gathered}{} \theta_{(k)}^0=\varrho h_{(k)}^0+\;\frac{1}{2}(\rho G_2)^{-1}\varrho \int_{s_1^0}^{s_1}\frac{1}{\varrho}F_{i(k-1)}^0\tau_i ds_1, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \eta_{(k)}^0=\varrho g_{(k)}^0+\;\frac{1}{2}(\rho G_2)^{-1}\varrho \int_{s_1^0}^{s_1}\frac{1}{\rho}F_{i(k-1)}^0\xi_i ds_1, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \theta_{(k)}^1=\rho h_{(k)}^1+\;\frac{1}{2}(\rho G_2)^{-1}\rho \int_{s_1^0}^{s_1}\frac{1}{\varrho}F_{i(k-1)}^1\tau_i ds_1, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \eta_{(k)}^1=\varrho g_{(k)}^1+\;\frac{1}{2}(\rho G_2)^{-1}\varrho \int_{s_1^0}^{s_1}\frac{1}{\varrho}F_{i(k-1)}^1\xi_i ds_1. \end{gathered} $$

The Ray Method and Its Applications

The discontinuities [Z,(k)] in the k-order partial time derivatives of a certain function Z obtained in the previous section are used thereafter for constructingthe ray series defining this desired function in the domain between the boundary of the thin body and the discontinuity surface Σ1 in the form (Rossikhin and Shitikova, 1994, 1995a)
$$\displaystyle \begin{aligned} Z&=\sum_{\alpha=1}^2\sum_{k=0}^\infty \frac{1}{k!}\; [Z,_{(k)}^{(\alpha)}]\Big|{}_{t=(s_1-s_1^0)/G_\alpha}\\&\quad \left(t-\frac{s_1-s_1^0}{G_\alpha}\right)^k H\left(t-\frac{s_1-s_1^0}{G_\alpha}\right), \end{aligned} $$
where the index α = 1, 2 denotes the ordinal number of the wave and H(t) is the unit Heaviside function.
At the boundary of the thin body, i.e., at \(s_1=s_1^0\), the ray series (52) takes the form
$$\displaystyle \begin{aligned} Z\Big|{}_{s_1=s_1^0}=\sum_{\alpha=1}^2\sum_{k=0}^\infty \frac{1}{k!}\; [Z,_{(k)}^{(\alpha)}]\Big|{}_{t=0}t^k. \end{aligned} $$
Suppose that on the boundary of the thin body, six functions to be found Z1, Z2, …, Z6 are given in the following form:
$$\displaystyle \begin{aligned} Z_i=\sum_{k=0}^\infty \frac{1}{k!}\;a_{ki}(s_2)t^k\quad (i=1,2,\ldots,6), \end{aligned} $$
where aki are unknown functions.
Then comparing relationships (53) for the desired functions Z1, Z2, …, Z6 with (54) yields
$$\displaystyle \begin{aligned} a_{ki}(s_2)&=\sum_{\alpha=1}^2[Z_{i,(k)}^{(\alpha)}]\Big|{}_{t=0}\\\quad &\qquad \quad (i=1,2,\ldots,6), (k=0,1,2,\ldots){} \end{aligned} $$

Equation (55) allows one to determine the arbitrary functions \(f_{(k)}^i(s_2)\), \(g_{(k)}^i(s_2)\), and \(h_{(k)}^i(s_2)\;\) (i = 0, 1) and thus to solve the boundary-value problem under consideration.

Below two examples of the ray series expansion utilization will be considered for solving the problems of a rigid washer pressing into a circular opening of an elastic plate and shallow cylindrical shell, which are of significant importance in mechanical engineering.

Pressing of a Rigid Washer into a Circular Opening of an Elastic Plate

Let a rigid washer of the mass m, radius R, and thickness H move with the speed V 0 toward an elastic isotropic plate of the thickness h with the circular opening of the radius R0 < R; in so doing the washer’s axis of symmetry coincides with the axis of symmetry for the opening (Fig. 2). The process of pressing of the washer into the plate begins at the moment t = 0, when the washer instantaneously enters the opening on the plate, and then begins to move relative to the plate vertically down with a variable speed. Besides, the Coulomb’s law of dry friction is fulfilled in the place of the washer and plate contact.

Starting from this moment, two cylindrical wave-strips appear in the plate; in so doing two circumferences L1 and L2 propagating along the plate’s median plane with the normal velocities G1 and G2 serve as the guides of the wave-strips. The character of the problem under consideration allows one to ignore flexural and torsional motions of the plate, as well as the in-plane shear deformations. Then proceeding from the above assumptions and considering that θ = π∕2, s1 = r, æ1 = 0, æ2 = −r−1, and \(\tau _1=\tau _2=\tau _2^g=0\), where r is the polar radius measured from the center of the opening, from the set of Eqs. (30)–(35) and (38)–(43), it follows, respectively, for the quasi-longitudinal wave
$$\displaystyle \begin{aligned} \frac{\delta\omega_{(k)}^0}{\delta r}+\;\frac{1}{2r}\;\omega_{(k)}^0 =\frac{1}{2\rho G_1}\;F_{i(k-1)}^0\lambda_i, \end{aligned} $$
$$\displaystyle \begin{aligned} \eta_{(k)}^0&=\frac{G_2^2G_1}{G_1^2-G_2^2} \left(-2\;\frac{\delta\eta_{(k-1)}^0}{\delta r} -\frac 1r\; \eta_{(k-1)}^0\right.\\ &\qquad ~\qquad \qquad +\left.\frac{G_1}{\rho G_2^2}\;F_{i(k-2)}^0\xi_i\right), \end{aligned} $$
Fig. 2

Scheme of the location of a rigid washer and a plate with a circular opening

and for the quasi-transverse wave
$$\displaystyle \begin{gathered}{} \frac{\delta\eta_{(k)}^0}{\delta r} +\frac{1}{2r}\;\eta_{(k)}^0 =\frac{1}{2\rho G_2}\;F_{i(k-1)}^0\xi_i, \end{gathered} $$
$$\displaystyle \begin{gathered} \omega_{(k)}^0=\frac{G_1^2G_2}{G_2^2-G_1^2} \left(-2\;\frac{\delta\omega_{(k-1)}^0}{\delta r} -\frac 1r\; \omega_{(k-1)}^0\right.\\ \left.\qquad \qquad \qquad \qquad + \frac{G_2}{\rho G_1^2}\;F_{i(k-2)}^0\lambda_i\right),{} \end{gathered} $$
Integrating recurrent equations (56) and (58) at k = 0, 1, 2, 3 and taking account for (52), one can obtain four-term ray expansions for the desired functions: \(v_r=\dot u_r=\omega =v_i\lambda _i\), W = η = viξi, δurδr, Nr = [σij]λiλj, and Qr = [σij]λiξj. These expansions have the following form:
$$\displaystyle \begin{aligned} v_r&=f_{(0)}^0r^{-1/2}+\left(f_{(1)}^0r^{-1/2} +\frac 38\;f_{(0)}^0G_1r^{-3/2}\right)\left(t-\frac{r-R}{G_1}\right)\\ &\quad +\frac 12\left(f_{(2)}^0r^{-1/2}+\frac 38\;f_{(1)}^0G_1r^{-3/2} -\frac{15}{128}\;f_{(0)}^0G_1^2r^{-5/2}\right)\left(t-\frac{r-R}{G_1}\right)^2\\ &\quad +\frac 16\left(f_{(3)}^0r^{-1/2}+\frac 38\;f_{(2)}^0G_1r^{-3/2}-\frac{15}{128}\;f_{(1)}^0G_1^2r^{-5/2} +\frac{105}{1024}\;f_{(0)}^0G_1^3r^{-7/2}\right)\\ &\quad \times \left(t-\frac{r-R}{G_1}\right)^3,\end{aligned} $$
$$\displaystyle \begin{aligned} W&=h_{(0)}^0r^{-1/2}+\left(h_{(1)}^0r^{-1/2}-\frac 18\;h_{(0)}^0G_2r^{-3/2}\right)\left(t-\frac{r-R}{G_2}\right)\\ &\quad +\frac 12\left(h_{(2)}^0r^{-1/2}-\frac 18\;h_{(1)}^0G_2r^{-3/2} +\frac{9}{128}\;h_{(0)}^0G_2^2r^{-5/2}\right)\left(t-\frac{r-R}{G_2}\right)^2\\ &\quad +\frac 16\left(h_{(3)}^0r^{-1/2}-\frac 18\;h_{(2)}^0G_2r^{-3/2}+\frac{9}{128}\;h_{(1)}^0G_2^2r^{-5/2} -\frac{75}{1024}\;h_{(0)}^0G_2^3r^{-7/2}\right)\\ &\quad \times \left(t-\frac{r-R}{G_2}\right)^3,\end{aligned} $$
$$\displaystyle \begin{aligned} G_1\;\frac{\partial u_r}{\partial r}&=-f_{(0)}^0r^{-1/2} +\left(-f_{(1)}^0r^{-1/2}-\frac 78\;f_{(0)}^0G_1r^{-3/2}\right)\left(t-\frac{r-R}{G_1}\right)\\ &\quad +\frac 12\left(-f_{(2)}^0r^{-1/2}-\frac 78\;f_{(1)}^0G_1 r^{-3/2}-\frac{57}{128}\;f_{(0)}^0G_1^2r^{-5/2}\right)\left(t-\frac{r-R}{G_1}\right)^2\\ &\quad +\frac 16\left(-f_{(3)}^0r^{-1/2}-\frac 78\;f_{(2)}^0G_1r^{-3/2}-\frac{57}{128}\;f_{(1)}^0G_1^2r^{-5/2} +\frac{195}{1024}\;f_{(0)}^0G_1^3r^{-7/2}\right)\\ &\quad \times \left(t-\frac{r-R}{G_1}\right)^3,\end{aligned} $$
$$\displaystyle \begin{aligned} N_r&=-\rho G_1hf_{(0)}^0r^{-1/2}+\rho h\left[-f_{(1)}^0G_1r^{-1/2} +\left(\frac 18\;G_1^2-2G_2^2\right)f_{(0)}^0r^{-3/2}\right]\left(t-\frac{r-R}{G_1}\right)\\ &\quad +\frac 12\;\rho h\left[-f_{(2)}^0G_1r^{-1/2}+\left(\frac 18\;G_1^2-2G_2^2\right)f_{(1)}^0r^{-3/2}\right.\\ &\quad -\left. \frac 38\;G_1\left(\frac{3}{16}\;G_1^2+2G_2^2\right)f_{(0)}^0r^{-5/2}\right]\left(t-\frac{r-R}{G_1}\right)^2\\ &\quad +\frac 16\;\rho h\left[-f_{(3)}^0G_1r^{-1/2}+\left(\frac 18\;G_1^2-2G_2^2\right)f_{(2)}^0r^{-3/2} -\frac 38\;G_1\left(\frac{3}{16}\;G_1^2+2G_2^2\right)f_{(1)}^0r^{-5/2}\right.\\ &\quad - \left. \frac{5}{128}\;G_1^2\left(\frac{71}{8}\;G_1^2-6G_2^2\right)f_{(0)}^0r^{-7/2}\right]\left(t-\frac{r-R}{G_1}\right)^3, \end{aligned} $$
$$\displaystyle \begin{aligned} Q_r&=-\rho G_2hh_{(0)}^0r^{-1/2}+\rho h\left(-h_{(1)}^0G_2r^{-1/2} -\frac 38\;h_{(0)}^0G_2^2r^{-3/2}\right)\left(t-\frac{r-R}{G_2}\right)\\ &\quad +\frac 12\;\rho h\left(-h_{(2)}^0G_2r^{-1/2}-\frac 38\;h_{(1)}^0G_2^2r^{-3/2} +\frac{15}{128}h_{(0)}^0G_2^3r^{-5/2}\right)\left(t-\frac{r-R}{G_2}\right)^2\\ &\quad +\frac 16\;\rho h\left(-h_{(3)}^0G_2r^{-1/2}-\frac 38\;h_{(2)}^0G_2^2r^{-3/2} +\frac{15}{128}\;h_{(1)}^0G_2^3r^{-5/2}\right.\\ &\quad - \left. \frac{105}{1024}\;h_{(0)}^0G_2^4r^{-7/2}\right)\left(t-\frac{r-R}{G_2}\right)^3, \end{aligned} $$
where \(f_{(i)}^0\) and \(h_{(i)}^0\) (i = 0, 1, 2, 3) are arbitrary constants.
Arbitrary constants are determined from the conditions of contact of the rigid washer with the plate
$$\displaystyle \begin{aligned} \frac{\partial u_r}{\partial r}\Bigg|{}_{r=R}=\frac{R_0-R}{R_0}=\varepsilon_r^0<0, \end{aligned} $$
$$\displaystyle \begin{aligned} fN_r=Q_r\Bigg|{}_{r=R}, \end{aligned} $$

where f is the coefficient of dry friction of the washer against the plate.

The difference in the washer’s diameter and the opening’s diameter is so small that the deformation \(\varepsilon _r^0\) is a purely elastic one. That is why the force of Coulomb’s friction is used for the force of resistance during the washer’s motion inside the opening.

Substituting (60)–(64) into (65) and (66) and equating the coefficients at like powers of t yield
$$\displaystyle \begin{gathered}{} f_{(0)}^0=-G_1R^{1/2}\varepsilon_r^0,\quad h_{(0)}^0=-G_1^2G_2^{-1}fR^{1/2}\varepsilon_r^0,\quad f_{(1)}^0=\frac 78\;G_1^2R^{-1/2}\varepsilon_r^0, \\ h_{(1)}^0=fG_1G_2\left(\frac{G_1^2}{G_2^2}+\frac 38\;\frac{G_1}{G_2}-2\right)R^{-1/2}\varepsilon_r^0,\quad f_{(2)}^0=-\frac{41}{128}\;G_1^3R^{-3/2}\varepsilon_r^0, \\ h_{(2)}^0=-\frac 12\;fG_1G_2^2\left(\frac{G_1^3}{G_2^3}+\frac 34\;\frac{G_1^2}{G_2^2} -\frac{95}{64}\;\frac{G_1}{G_2}-\frac 32\right)R^{-3/2}\varepsilon_r^0,\quad f_{(3)}^0=-\frac{307}{1024}\;G_1^4R^{-5/2}\varepsilon_r^0, \\ h_{(3)}^0=\frac{1}{16}\;fG_1G_2^3\left(-\frac{279}{32}\;\frac{G_1^4}{G_2^4} +3\frac{G_1^3}{G_2^3}+\frac{23}{8}\;\frac{G_1^2}{G_2^2}-\frac{135}{64}\;\frac{G_1}{G_2} -\frac{33}{4}\right)R^{-5/2}\varepsilon_r^0. \end{gathered} $$
Substituting (67) into the four-term truncated ray series (60)–(64), the desired values vr, W, Nr, and Qr could be found. Knowing the t-dependence of the value Nr, it is possible to obtain the law of washer’s motion relative to the plate and, among other factors, to determine the time of the washer’s motion and the length of the pathway traversed by the washer during its motion. For this purpose, let us write the equation of motion of the rigid washer
$$\displaystyle \begin{aligned} ma=-f|N_r|, \end{aligned} $$
where \(a=\dot v\) is the acceleration of the rigid washer.
The initial condition
$$\displaystyle \begin{aligned} v\Bigg|{}_{t=0}=V_0 \end{aligned} $$
should be added to (68).
Substituting (63) into Eq. (68) with due account for (67) and limiting by the terms of the order of t yield
$$\displaystyle \begin{aligned} \frac{dv}{dt}=A+Bt, \end{aligned} $$
$$\displaystyle \begin{aligned} A&=fG_1^2\;\frac{\rho h}{m}\;\varepsilon_r^0<0, \\ B&=-fG_{(1)}^3\;\frac{\rho h\sigma}{mR}\;\varepsilon_r^0>0. \end{aligned} $$
Integrating (70) and taking the initial condition (69) into consideration yield
$$\displaystyle \begin{aligned} v=V_0+At+B\;\frac{t^2}{2}. \end{aligned} $$
Putting v = 0, it is possible to find the time
$$\displaystyle \begin{aligned} t_0=-\;\frac{A}{B}-\sqrt{\frac{A^2}{B^2}-\frac{2V_0}{B}} \end{aligned} $$
during which the speeds of the washer and the points possessed by the plate opening become equal to each other, i.e., the washer ceases to move relative to the plate. In Eq. (72) before the square root, the sign “minus” is chosen, since at V 0 → 0, the time t0 should tend to zero.
Integrating (71) with respect to the time t and putting t = t0 in the net expression, the distance traveled by the washer relative to the plate could be found
$$\displaystyle \begin{aligned} u=V_0\;t_0+A\;\frac{t_0^2}{2}+B\;\frac{t_0^3}{6}. \end{aligned} $$

The values defined by (71)–(73) completely characterize the process of pressing of the washer beginning with the moment of its impact upon the plate and ending with its stopping. The analysis of the expressions (71)–(73) allows one to find the initial speed V 0 of the washer at the moment of impact such that the washer will occupy the position relative to the plate required by the process conditions.

Pressing of a Rigid Circular Washer into an Opening in a Lateral Surface of an Elastic Shallow Cylindrical Shell

Let a rigid washer of the mass m, radius R, and thickness H move with the speed V 0 normally to a lateral surface of an elastic isotropic shallow cylindrical shell of the radius a and thickness h with the circular opening of the radius R0 < R in its lateral surface; in so doing the washer’s axis of symmetry coincides with the axis of symmetry of the opening. The process of pressing of the washer into the shell begins at the moment t = 0, when the washer instantaneously enters the opening on the shell lateral surface, and then begins to move relatively to the shell along the normal to its middle surface with a variable speed. Besides, the Coulomb’s law of dry friction is fulfilled in the place of the washer and shell contact.

From the moment of impact, disturbances propagate along the rays, which are the geodetic lines on the cylindrical surface, i.e., helical lines. Along these lines
$$\displaystyle \begin{aligned} \text{\ae}_1&=\frac{a}{a^2+b^2},\quad \tau_1=\frac{b}{a^2+b^2},\\\quad \text{\ae}_2&=-\;\frac{1}{s_1},\quad \tau_2^g=-\tau_1,{} \end{aligned} $$
where b = h∕2π and h is the lead of a circular helix.
If one develops a cylindrical surface on a plane (Fig. 3), then all geometrical parameters of the family of helical lines issuing from the center of the opening can easily be expressed in terms of the angle of inclination φ of each helix to the horizontal axis. Really, starting from the notation of a helical line
$$\displaystyle \begin{aligned} \tan\gamma=\frac ab, \end{aligned}$$
and hence
$$\displaystyle \begin{aligned} \tan\varphi=\frac ba,\quad \text{\ae}_1=\frac{\cos^2\varphi}{a},\quad \tau_1=\frac{\sin2\varphi}{a}. \end{aligned} $$
Fig. 3

Development of a cylindrical surface with an opening on a plane

For further calculations, it is a need to express \(\sin \theta \) and \(\cos \theta \) in terms of φ and s1 as follows (Rossikhin and Shitikova, 2007)
$$\displaystyle \begin{aligned} \cos\theta&=-\;\frac{s_1}{a}\;\sin^2\varphi\Bigg[\frac{1}{2} \sin\left(2\;\frac{s_1}{a}\;\cos\varphi\right)\\ &\qquad \qquad \qquad \quad +\sin^2\left(\frac{s_1}{a}\;\cos\varphi\right)\Bigg],\\ &\qquad \qquad \qquad \qquad\sin\theta=1,{} \end{aligned} $$
wherein the angle φ remains constant along each helix from the family under consideration and changes its magnitude during transition from the one line of the family to the other, i.e., it is a parameter of this family of helical lines.
The behavior of two wave surfaces in the shell is described by the set of Eqs. (30)–(35) and (38)–(43), wherein relationships (74) and (75) should be taken into account, whence it follows on the quasi-longitudinal wave
$$\displaystyle \begin{gathered}{} \frac{\delta\omega_{(k)}^0}{\delta s_1}+\;\frac{1}{2s_1}\;\omega_{(k)}^0 =\frac{1}{2\rho G_1}\;F_{i(k-1)}^0\lambda_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \theta_{(k)}^0=-\;\frac{G_2^2G_1}{G_1^2-G_2^2} \left(2\;\frac{\delta\theta_{(k-1)}^0}{\delta s_1} + \frac{1}{s_1}\; \theta_{(k-1)}^0 -\frac{3}{2a}\;\sin2\varphi\;\eta_{(k-1)}^0 -\frac{G_1}{\rho G_2^2}\;F_{i(k-2)}^0\tau_i\right), \end{gathered} $$
$$\displaystyle \begin{gathered}{} \eta_{(k)}^0=-\;\frac{G_2^2G_1}{G_1^2-G_2^2} \left(2\;\frac{\delta\eta_{(k-1)}^0}{\delta s_1} + \frac{1}{s_1}\; \eta_{(k-1)}^0 +\frac{3}{2a}\;\sin2\varphi\;\theta_{(k-1)}^0 -\frac{G_1}{\rho G_2^2}\;F_{i(k-2)}^0\xi_i\right), \end{gathered} $$
and for the quasi-transverse wave
$$\displaystyle \begin{gathered}{} \frac{\delta\theta_{(k)}^0}{\delta s_1} +\frac{1}{2s_1}\;\theta_{(k)}^0 -\frac{3}{4a}\;\sin2\varphi\;\eta_{(k)}^0 =\frac{1}{2\rho G_2}\;F_{i(k-1)}^0\tau_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \frac{\delta\eta_{(k)}^0}{\delta s_1} +\frac{1}{2s_1}\;\eta_{(k)}^0 +\frac{3}{4a}\;\sin2\varphi\;\theta_{(k)}^0 =\frac{1}{2\rho G_2}\;F_{i(k-1)}^0\xi_i, \end{gathered} $$
$$\displaystyle \begin{gathered}{} \omega_{(k)}^0=\frac{G_1^2G_2}{G_1^2-G_2^2} \left(2\;\frac{\delta\omega_{(k-1)}^0}{\delta s_1} +\frac{1}{s_1}\; \omega_{(k-1)}^0 - \frac{G_2}{\rho G_1^2}\;F_{i(k-2)}^0\lambda_i\right), \end{gathered} $$
where functions \(F_{i(k-1)}^0\lambda _i\), \(F_{i(k-1)}^0\xi _i\), and \(F_{i(k-1)}^0\tau _i\) depend on the (k − 1)-order discontinuities of the desired values (Rossikhin and Shitikova, 2007).
Putting k = 0,  1,  2,  …  in Eqs. (77)–(79) and (80)–(82) yields on the quasi-longitudinal wave
$$\displaystyle \begin{aligned} \omega_{(0)}^0 &= f_{(0)}(\varphi)s_1^{-1/2},\qquad\theta_{(0)}^0=\eta_{(0)}^0=0, \\ \theta_{(1)}^0 &= -G_1f_{(0),\varphi}s_1^{-3/2},\qquad\eta_{(1)}^0=\frac{G_1(G_1^2+G_2^2)}{(G_1^2-G_2^2)a}\;\cos^2\varphi f_{(0)}s_1^{-1/2}, \\ \omega_{(1)}^0 &= f_{(1)}(\varphi)s_1^{-1/2} +\frac{1}{2}\;G_1\left(\frac{G_1^2-2G_2^2}{G_1^2}\;f_{(0),\varphi\varphi} +\frac 34\;f_{(0)}\right)s_1^{-3/2}\\ &\quad - \frac{1}{2} \;G_1\left(\frac{G_2^2}{4G_1^2}\;\sin^22\varphi + \frac{G_1^2+5G_2^2}{G_1^2-G_2^2}\;\cos^4\varphi \right)f_{(0)}\;\frac{s_1^{1/2}}{a^2}, \ldots\end{aligned} $$
and on the quasi-transverse wave
$$\displaystyle \begin{aligned} \theta_{(0)}^0 &= h_{(0)}(\varphi)s_1^{-1/2}+\frac{3}{4a}\;\sin2\varphi\; g_{(0)}(\varphi)s_1^{1/2}, \\ \eta_{(0)}^0 &= g_{(0)}(\varphi)s_1^{-1/2}-\frac{3}{4a}\;\sin2\varphi\; h_{(0)}(\varphi)s_1^{1/2},\quad \omega_{(0)}^0=0, \\ \omega_{(1)}^0 &= G_2\left(h_{(0),\varphi}s_1^{-3/2}+\frac{3}{4a}\;\sin2\varphi\; g_{(0),\varphi}s_1^{-1/2} +\frac{3}{2a}\;\cos2\varphi\; g_{(0)}s_1^{-1/2}\right)\\ &\quad + \frac{G_2(G_1^2+2G_2^2)}{G_1^2-G_2^2}\;\frac{\cos^2\varphi}{a} \left(g_{(0)}s_1^{-1/2}-\frac{3}{4a}\;\sin2\varphi\; h_{(0)}s_1^{-1/2}\right),\ldots\end{aligned} $$
where f(i)(φ), g(i)(φ), and h(i)(φ) (i = 0, 1) are arbitrary functions to be determined from the conditions of contact of the rigid washer with the shell, which are valid within the contact region and on its boundary:
$$\displaystyle \begin{aligned} \frac{\partial u_\lambda}{\partial s_1}\Bigg|{}_{s_1=R}=\frac{R_0-R}{R_0} =\varepsilon_\lambda^0<0, \end{aligned} $$
$$\displaystyle \begin{aligned} fN_\lambda=Q_\lambda\Big|{}_{s_1=R}, \end{aligned} $$
$$\displaystyle \begin{aligned} v_\tau\Big|{}_{s_1=R}=0, \end{aligned} $$
and f is the coefficient of dry friction of the washer against the shell.
Limiting by the discontinuities in the displacement velocities of the order k = 0, 1, the discontinuities in the stress tensor components (12)–(16) of the same order can be calculated and specifically the values of [σλλ(k)] and [σλξ(k)] at k = 0, 1:
$$\displaystyle \begin{aligned} \left[\sigma_{\lambda\lambda (0)}^{(1)}\right] &= -\rho G_1\;\omega_{(0)}^{0(1)}, \qquad\left[\sigma_{\lambda\lambda (0)}^{(2)}\right]=0, \\ \left[\sigma_{\lambda\lambda (1)}^{(1)}\right] &= -\rho G_1\;\omega_{(1)}^{0(1)} +\rho G_1^2\;\frac{\delta\omega_{(0)}^{0(1)}}{\delta s_1} +(\rho G_1^2-2\rho G_2^2)\;\frac{1}{s_1}\;\omega_{(0)}^{0(1)},\\ \left[\sigma_{\lambda\lambda (1)}^{(2)} \right] &= -\;\frac{\rho G_1^2}{G_2}\;\omega_{(1)}^{0(2)} +\rho G_1^2\;\frac{\cos^2\varphi}{a}\;\eta_{(0)}^{0(2)} +(\rho G_1^2-2\rho G_2^2)\;\frac{1}{s_1}\; \frac{\delta\theta_{(0)}^{0(2)}}{\delta\varphi};\end{aligned} $$
$$\displaystyle \begin{aligned} \left[\sigma_{\lambda\xi(0)}^{(1)}\right] &=0, \quad \left[\sigma_{\lambda\xi(0)}^{(2)}\right]=-\rho G_2\;\eta_{(0)}^{0(2)}, \\ {}\left[\sigma_{\lambda\xi(1)}^{(1)}\right] &=-\rho G_2^2\;\frac{2G_1^2}{\rho G_1^2-\rho G_2^2} \frac{\cos^2\varphi}{a}\;\omega_{(0)}^{0(1)},\\ {}\left[\sigma_{\lambda\xi(1)}^{(2)}\right] &=-\rho G_2\;\eta_{(1)}^{0(2)}\\ {}&\quad +\rho G_2^2\left(\frac{\delta\eta_{(0)}^{0(2)}}{\delta s_1} +\frac{\sin2\varphi}{2a}\;\theta_{(0)}^{0(2)}\right),\end{aligned} $$
which, in its turn, allow one to construct the two-term truncated ray series for the values vξ = viξi, vλ = viλi, vθ = viτi, σλλ, and σλξ,
$$\displaystyle \begin{aligned} v_\xi&=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\eta_{(k)}^{(\alpha)}\\ {}&\quad \times\left(t-\frac{s_1-R}{G_\alpha}\right)^k H\left(t-\frac{s_1-R}{G_\alpha}\right), \end{aligned} $$
$$\displaystyle \begin{aligned} v_\lambda&=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\omega_{(k)}^{(\alpha)}\\ {}&\quad \times\left(t-\frac{s_1-R}{G_\alpha}\right)^k H\left(t-\frac{s_1-R}{G_\alpha}\right), \end{aligned} $$
$$\displaystyle \begin{aligned} v_\theta&=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\theta_{(k)}^{(\alpha)}\\ {}&\quad \times\left(t-\frac{s_1-R}{G_\alpha}\right)^k H\left(t-\frac{s_1-R}{G_\alpha}\right), \end{aligned} $$
$$\displaystyle \begin{aligned} \sigma_{\lambda\lambda}&=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\; [\sigma_{\lambda\lambda(k)}^{(\alpha)}]\\ {}&\quad \times\left(t-\frac{s_1-R}{G_\alpha}\right)^k H\left(t-\frac{s_1-R}{G_\alpha}\right), \end{aligned} $$
$$\displaystyle \begin{aligned} \sigma_{\lambda\xi}&=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\; [\sigma_{\lambda\xi(k)}^{(\alpha)}]\\ {}&\quad \times\left(t-\frac{s_1-R}{G_\alpha}\right)^k H\left(t-\frac{s_1-R}{G_\alpha}\right), \end{aligned} $$
where the ray series coefficients are calculated at t = (s1 − R)∕Gα.
On the boundary of the contact region, i.e., at s1 = R, the above truncated ray series take the form
$$\displaystyle \begin{aligned} &\widetilde{v_\xi}=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\eta_{(k)}^{(\alpha)}\Big|{}_{t=0}t^k, \end{aligned} $$
$$\displaystyle \begin{aligned} &\widetilde{v_\lambda}=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\omega_{(k)}^{(\alpha)}\Big|{}_{t=0}t^k, \\\quad &\widetilde{v_\tau}=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\;\theta_{(k)}^{(\alpha)}\Big|{}_{t=0}t^k, {} \end{aligned} $$
$$\displaystyle \begin{aligned} &\widetilde{\sigma_{\lambda\lambda}}=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\; [\sigma_{\lambda\lambda(k)}^{(\alpha)}]\Big|{}_{t=0}t^k,\\\quad &\widetilde{\sigma_{\lambda\xi}}=\sum_{\alpha=1}^2\sum_{k=0}^1\frac{1}{k!}\; [\sigma_{\lambda\xi(k)}^{(\alpha)}]\Big|{}_{t=0}t^k,{} \end{aligned} $$
where a tilde over the values denotes that the given value is calculated on the boundary of the contact spot.
The ray expansions (95)-(97) allow one to investigate the motion of a rigid washer inside an opening in the lateral surface of the cylindrical shell. The equation of motion of the rigid washer subjected to the initial condition (69) could be written as
$$\displaystyle \begin{aligned} m\;\frac{dv}{dt}=hR\int_0^{2\pi}\sigma_{\lambda\xi}d\varphi, \end{aligned} $$
which within an accuracy of the terms of the order of t is reduced to Eq. (70) where
$$\displaystyle \begin{aligned} &A=2\pi fG_1^2\;\frac{\rho hR}{m}\;\varepsilon_r^0<0,\\ \quad &B=-\;\frac{\pi fG_{(1)}^4\rho hR}{maG_{(2)}}\\ &\qquad \times\left[f+\frac{2G_{(2)}R}{G_{(1)}a}\left(1-\;\frac{2G_{(2)}^2}{G_{(1)}^2} \right) \right]\varepsilon_r^0>0. \end{aligned} $$

Thus, further reasoning is similar to that of the analysis of Eq. (70).

Reference to the relationship for the coefficient B in (99) shows that the duration of the washer’s motion within the shell’s opening decreases with increasing shell curvature. When a →, the cylindrical shell degenerates into a plate, and relationships (99) for the coefficients A and B go over into the expressions involved in Eq. (70).

Thus, in the examples considered above, two types of the rays are met: straight lines and helical lines (helical rays). During the transient shear wave propagation along the straight rays, the displacement velocity vector on this wave remains unchanged both in magnitude and in direction, i.e., polarization of the wave is constant with time. During the transient shear wave propagation along the rays possessing curvature and torsion, as in the case of helical lines, the velocity vector changes both in magnitude and in direction, i.e., the wave polarization varies as time goes on.

That is why a shear wave that splits into two wave fronts propagating with different velocities, on one of which shear occurs along the middle surface whereas on the other one shear takes place transversely to the middle surface, may be justified only for the straight and plane rays. For the spatial rays, such splitting of the shear wave is impermissible, since polarization of this wave depends on the distance travelled along the ray. Thus, the dynamic equations of the Timoshenko type describing the wave behavior of different thin-walled bodies are of limited application.


The theory of transient dynamic behavior of thin bodies has been constructed which is based on the 3D equations of the theory of elasticity, the theory of discontinuities, and the ray method. The foundation of this theory is two assumptions: (1) the wave surface-strip is always perpendicular to the median surface of the thin body, and (2) the stresses normal to the median surface of the thin body do not experience discontinuities during transition through the wave surface strip.

The theory proposed allows one to investigate the peculiarities of the dynamic behavior of thin bodies under short-time loading of those bodies, which is characteristic to the problems connected with the shock interaction, dynamic stability, fracture, and so on (e.g., see entries “Ray Expansions in Impact Interaction Problems,” “Ray Expansions in Dynamic Contact Problems,” and “Collision of Two Spherical Shells, Fractional Operator Models”).

Unlike other dynamic theories of plates and shells based on the 3D equations of the theory of elasticity, the present theory possesses simplicity and clearness and does not involve new material constants introduced by artificial means.



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

Authors and Affiliations

  1. 1.Research Center on 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