# Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

# Ray Expansions in Impact Interaction Problems

Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_99-1

## Definitions

The impact interaction problems are those connected with the shock interactions of bodies subjected to the contact law.

## Preliminary Remarks

The problems connected with the analysis of the shock interaction of thin bodies (rods, beams, plates, and shells) with other bodies have widespread application in various fields of science and technology. The physical phenomena involved in the impact event include structural responses, contact effects, and wave propagation. These problems are topical not only from the point of view of fundamental research in applied mechanics but also with respect to their applications. Because these problems belong to the problems of dynamic contact interaction, their solution is connected with severe mathematical and calculation difficulties. To overcome this impediment, a rich variety of approaches and methods have been suggested, and the overview of current results in the field can be found in state-of-the-art articles by Abrate (2001) and Rossikhin and Shitikova (2007a).

In many engineering applications, it is important to understand the transient behavior of isotropic as well as composite thin-walled shell structures subjected to central impact by a small projectile. Recently Rossikhin and Shitikova (2007b) have developed a new formulation of the ray method which is applicable for analyzing the propagation of surfaces of strong and weak discontinuity in thin elastic bodies when the wave fronts and the rays are referenced to the curvilinear system of coordinates (Rossikhin and Shitikova, 1995b). It should be noted that the ray method is primarily used for obtaining the problem solution analytically. This approach is based on the reduction of the three-dimensional equations of the dynamic theory of elasticity, which first should be written in discontinuities, to the two-dimensional equations by virtue of integration over the coordinate perpendicular to the middle surface of a thin body. The recurrent equations of this ray method are free from the shear coefficient, which is usually inherent to the Timoshenko-type theories (Timoshenko, 1936), and involve only two elastic constants: Poisson’s ratio and elastic modulus of elongation.

The theory proposed in Rossikhin and Shitikova (2007b) is applicable for short times after the passage of the wave front, but it possesses the simplicity inherent in the “classical” theory of thin bodies. The advantages of this approach will be illustrated in this entry by solving the engineering problems on normal impact of an elastic spherical and long cylindrical hemisphere-nose projectiles against an elastic spherical shell using the nonlinear Hertzian law within the contact region.

## Impact Response of a Spherical Shell of the Timoshenko Type

Let an elastic sphere with the radius r0 and mass m (Fig. 1) or a long cylindrical elastic rod of radius r0 with a hemispherical nose of the same radius (Fig. 2) move along the x3-axis with the velocity V 0 towards an elastic isotropic spherical shell of the R radius (Rossikhin et al., 2011). Fig. 1 Scheme of the shock interaction of a falling sphere with a spherical shell Fig. 2 Scheme of the shock interaction of a falling cylindrical rod with a shell

The impact occurs at the initial instant of time at x3 = R. At the moment of impact, two shock wave lines (surfaces of strong discontinuity) are generated in the shell, which then propagate along the shell during the process of impact. During transition through the wave line, the following wave fields experience the discontinuities: stresses, velocities of displacements, and the values of the higher-order time derivatives in the displacements.

### Geometry of the Wave Surface

A wave-strip is a ruled cylindrical surface consisting of the directrix C, which is the wave line propagating along the median surface of the shell, and the family of generatrices representing the line segments of the length h, which are perpendicular to the shell’s median surface and thus to the wave line, and which are fitted to the wave line by their middles. Let us take the family of generatrices as the u1-curves, where u1 is the distance measured along the straight line segment from the C curve, and choose the distance measured along the C curve as u2 (Fig. 3). The u1-family is the family of geodetic lines. In this case, all conditions of the McConnel theorem are fulfilled, and a linear element of the wave surface takes the form (McConnel, 1957)
\displaystyle \begin{aligned} ds^2=(du^1)^2+g_{22}(u^1,u^2)(du^2)^2, \end{aligned}
(1)
in so doing
\displaystyle \begin{aligned} g_{22}(0,u^2)=1, \end{aligned}
(2)
where g11 = 1, g22, and g12 = 0 are the covariant components of the metric tensor of the wave surface. Fig. 3 Scheme of the propagating wave-strip along the spherical shell surface
The Gaussian curvature for the linear element (1) is defined by the following formula (McConnel, 1957):
\displaystyle \begin{aligned} K=-\frac{1}{\sqrt{g_{22}}}\;\frac{\partial ^2 \sqrt{g_{22}}}{(\partial u^1)^2}=0. \end{aligned}
(3)
Integrating Eq. (3) and considering formula (2) yield
\displaystyle \begin{aligned} \sqrt{g_{22}}=1+c u^1, \end{aligned}
(4)
where c is a certain constant.
It is known that small distances along the coordinate lines u2 are defined by the formula (McConnel, 1957)
\displaystyle \begin{aligned} ds_2=\sqrt{g_{22}} \;d u^2, \end{aligned}
or considering (4)
\displaystyle \begin{aligned} ds_2=(1+c u^1) d u^2. \end{aligned}
(5)
Rewriting formula (5) in the form
\displaystyle \begin{aligned} \frac{d s_2-d u^2}{d u^2}=c u^1, \end{aligned}
and integrating the result relationship with respect to u1 from − h∕2 to h∕2 yield
\displaystyle \begin{aligned} \int_{-h/2}^{h/2}\frac{d s_2-d u^2}{d u^2} \;d u^1=0, \end{aligned}
or
\displaystyle \begin{aligned} \int_{-h/2}^{h/2}\frac{ds_2}{d u^2} \;d u^1=h. \end{aligned}
(6)
Equation (6) can be written as
\displaystyle \begin{aligned} \frac{1}{h}\;\int_{-h/2}^{h/2} \sqrt{g_{22}} \;d u^1=1, \end{aligned}
i.e., the mean magnitude of the value $$\sqrt {g_{22}}$$ over the thickness of the shell is equal to unit.
If the shell’s thickness is small, then it is possible to consider approximately that
\displaystyle \begin{aligned} \sqrt{g_{22}} \approx 1 \end{aligned}
(7)
at any point of the wave surface. Since all values for the shell are averaged over its thickness, then such an approximation for $$\sqrt {g_{22}}$$ is not unreasonable.
The linear element (1) with due account for (7) can be approximately written as
\displaystyle \begin{aligned} d s^2 \approx (d u^1)^2+ (d u^2)^2, \end{aligned}
(8)
i.e., it looks like a linear element on the plane in the Cartesian set of coordinates.
Now let us define a linear element of the median surface of the shell. Since the rays intersecting the line C (the wave line) under the right angles are the family of the geodetic lines, then once again the conditions of the McConnel theorem remain valid, and thus the linear element of this surface takes the form
\displaystyle \begin{aligned} d s^2 = (d u_*^1)^2+g_{22} (d u^2)^2, \end{aligned}
(9)
but considering formula (7), it can be rewritten in the form of (8) by substituting du1 by $$du_*^1$$.

### The Main Kinematic and Dynamic Characteristics of the Wave Surface

The condition of compatibility on the wave surface of strong discontinuity in view of (7), (8) and (9) takes the form (see for details the entry “Ray Expansion Theory”)
\displaystyle \begin{aligned} \begin{array}{rcl} {} [u_{i,j(k)}]&\displaystyle =&\displaystyle - G^{-1} [v_{i,(k)}]\lambda_j +\frac{\delta [u_{i,(k)}]}{\delta s_1}\;\lambda _j \\ {} &\displaystyle +&\displaystyle \frac{\delta [u_{i,(k)}]}{\delta s_2}\;\tau _j +\left[\frac{\delta u_{i,(k)} \xi _j}{\delta \xi } \right], \end{array} \end{aligned}
(10)
where ui are the displacement vector components; G is the normal velocity of the wave surface; [ui,j] = [∂ui∂xj]; xj are the spatial rectangular Cartesian coordinates; ξ = u1; $$s_1=u_*^1$$; [ui,(k)] = [kui∂tk]; t is the time; vi = ui,(1); λi, τi, and ξi are the components of the unit vectors of the tangential to the ray, the tangential to the wave surface, and the normal to the spherical surface, respectively; and Latin indices take on the values 1,2,3.
Putting k = 0 in (10) yields
\displaystyle \begin{aligned}{}[u_{i,j}]=-G^{-1} [v_i]\lambda _j+\left[\frac{\delta (u_{i}\xi _j )}{\delta \xi } \right]. \end{aligned}
(11)
Writing the Hooke’s law for a three-dimensional medium in terms of discontinuities and using the condition of compatibility (11) yield
\displaystyle \begin{aligned}{}[\sigma _{ij}]&=-G^{-1}\lambda [v_\lambda ]\delta _{ij}-G^{-1}\mu \left( [v_i]\lambda _j +[v_j]\lambda _i\right) \\ &\quad +\lambda [u_{\xi ,\xi }]\delta _{ij}+\mu \left(\left[\frac{\delta (u_{i}\xi _j )}{\delta \xi } +\frac{\delta (u_{j}\xi _i )}{\delta \xi } \right]\right)\ \ \quad \end{aligned}
(12)
where
\displaystyle \begin{aligned}{}[v_\lambda ]=[v_i]\lambda _i,\quad [u_{\xi ,\xi }]=\left[\frac{\delta (u_{i}\xi _i)}{\delta \xi }\right]= \left[\frac{\delta u_\xi }{\delta \xi }\right], \end{aligned}
λ and μ are Lame constants, and δij is the Kronecker’s symbol.
Multiplying relationship (12) from the right and from the left by ξiξj and considering equation
\displaystyle \begin{aligned}{}[\sigma _{\xi \xi }]=[\sigma _{ij}]\xi _i\xi _j=0, \end{aligned}
what corresponds to the assumption that the normal stresses on the cross-sections parallel to the middle surface could be neglected with respect to other stresses, it could be found
\displaystyle \begin{aligned}{}[u _{\xi, \xi }]=\frac{\lambda }{G(\lambda +2\mu )}\;[v _\lambda ]. \end{aligned}
(13)
Multiplying relationship (12) from the right and from the left by λiλj leads to the equation
\displaystyle \begin{aligned}{}[\sigma _{\lambda \lambda }]=[\sigma _{ij}]\lambda _i\lambda _j =-G^{-1}(\lambda +2\mu ) [v_\lambda ]+\lambda [u_{\xi ,\xi }]. \end{aligned}
(14)
Substituting (13) in (14) yields
\displaystyle \begin{aligned}{}[\sigma _{\lambda \lambda }]=-\frac{4\mu (\lambda +\mu )}{\lambda +2\mu }\; G^{-1}[v_\lambda ], \end{aligned}
or
\displaystyle \begin{aligned}{}[\sigma _{\lambda \lambda }]=-\frac{E}{1-\sigma ^2}\; G^{-1}[v_\lambda ], \end{aligned}
(15)
where E and σ are the elastic modulus and the Poisson’s ratio, respectively.
Alternatively, multiplying the three-dimensional equation of motion written in terms of discontinuities
\displaystyle \begin{aligned}{}[\sigma _{ij }]\lambda _j=-\rho G [v_i ], \end{aligned}
(16)
by λi yields
\displaystyle \begin{aligned}{}[\sigma _{\lambda \lambda }]=-\rho G [v_\lambda ], \end{aligned}
(17)
where ρ is the density of the shell’s material.
Eliminating the value [σλλ] from (15) and (17), the velocity of the quasi-longitudinal wave propagating in the spherical shell is defined as
\displaystyle \begin{aligned} G_1=\sqrt{\frac{E}{\rho (1-\sigma ^2)}}. \end{aligned}
(18)
Relationship (15) with due account for (18) takes the form
\displaystyle \begin{aligned}{}[\sigma _{\lambda \lambda }]=-\rho G_1 [v_\lambda ]. \end{aligned}
(19)
Multiplying (12) by λiξj and (16) by ξi yields
\displaystyle \begin{aligned}{}[\sigma _{\lambda \xi }]=[\sigma _{ij}]\lambda _i\xi _j= -\mu G^{-1} [v_\xi ], \end{aligned}
(20)
\displaystyle \begin{aligned}{}[\sigma _{\lambda \xi }]=-\rho G [v_\xi ], \end{aligned}
(21)
where [vξ] = [vi]ξi.
Eliminating the value [σλξ] from (20) and (21), the velocity of the quasi-transverse wave is defined as
\displaystyle \begin{aligned} G_2=\sqrt{\frac{\mu }{\rho}}. \end{aligned}
(22)
Considering (22), relationship (20) takes the form
\displaystyle \begin{aligned}{}[\sigma _{\lambda \xi }]=-\rho G_2 [v_\xi ]. \end{aligned}
(23)
Note that in the three-dimensional medium, only one value, i.e., [uλ,λ], is nonzero on the quasi-longitudinal wave, while in the two-dimensional medium, where the “wave-strip” propagates, on the quasi-longitudinal wave, there are two nonvanishing values, namely, [uλ,λ] and [uξ,ξ]. Between these two values, it is possible to find the relationship. For this purpose, let us multiply (11) from right and from left by λiλj and express the values [vλ]
\displaystyle \begin{aligned}{}[v_\lambda ]=-G_1[u_{\lambda ,\lambda }], \end{aligned}
and then the obtained expression should be substituted in (13). As a result the desired linkage could be found
\displaystyle \begin{aligned}{}[u _{\xi ,\xi }]=-\frac{\sigma }{1-\sigma }\; [u_{\lambda ,\lambda } ]. \end{aligned}
(24)
However, if one considers the strains in a thin body, for example, a plate in the rectangular Cartesian set of coordinates, assuming that
\displaystyle \begin{aligned} \sigma _{zz}=\frac{E\left[ (1-\sigma ) u_{z,z}+\sigma (u_{x,x}+u_{y,y})\right]} {(1+\sigma )(1-2\sigma )} =0, \end{aligned}
then it is possible to obtain a little bit another formula
\displaystyle \begin{aligned} u _{z,z}=-\frac{\sigma }{1-\sigma }\; (u_{x,x}+u_{y,y}). \end{aligned}
(25)

From the comparison of (24) and (25), it is seen that in the right-hand side of (24), the value [uτ,τ] = [uij]τiτj is absent, but its absence is connected with the peculiarities of the “wave-strip,” namely, it has free edges at ξ = ±h∕2 and a closed contour with respect to s2.

### Governing Equations

Thus, behind the front of each of two transient waves (surfaces of strong discontinuity) up to the boundary of the contact domain (Figs. 1 or 2), relationships (19) and (23) are valid, which are the first terms of the ray expansions (Fig. 4), i.e.,
\displaystyle \begin{aligned} \sigma _{\lambda \lambda }&=-\rho G_1 v_\lambda , \end{aligned}
(26)
\displaystyle \begin{aligned} \sigma _{\lambda \xi }&=-\rho G_2 v_\xi . \end{aligned}
(27) Fig. 4 Scheme of velocities and stresses in the shell’s element on the boundary of the contact domain
Considering the cone angle of the contact spot 2γ as a small value (Fig. 4), and putting $$\cos \gamma {\approx } 1$$, $$\sin \gamma \approx \gamma =aR^{-1}$$ yield
\displaystyle \begin{aligned} \widetilde{v_z}&=\widetilde{v_\xi } -\widetilde{v_\lambda }\frac{a}{R}, \end{aligned}
(28)
\displaystyle \begin{aligned} \widetilde{v_r}&=\widetilde{v_\xi }\frac{a}{R} +\widetilde{v_\lambda }, \end{aligned}
(29)
\displaystyle \begin{aligned} \widetilde{\sigma _{rz}}&=\rho G_1 \widetilde{v_\lambda }\frac{a}{R}-\rho G_2 \widetilde{v_\xi }, \end{aligned}
(30)
where a is the radius of the contact spot and $$\widetilde {\sigma _{rz}}=\sigma _{rz}|{ }_{r=a}$$.
According to the Hertzian theory of contact, during the loading phase, the contact force Fcont is related to the indentation α (i.e., the difference between the displacements of impactor and target or the local bearing of impactor and target materials), by the relationship
\displaystyle \begin{aligned} F_{\mathrm{cont}}=k\alpha ^{3/2}, \end{aligned}
(31)
where k is the contact stiffness coefficient depending on the geometry of colliding bodies, as well as their elastic constants:
\displaystyle \begin{aligned} k&=\frac{4}{3\pi }\;\frac{\sqrt{R\prime}}{k\prime+k\prime\prime}, \quad k\prime=\frac{1-\sigma ^2}{E},\\ k\prime\prime&=\frac{1-\sigma_{\mathrm{im}} ^2}{E_{\mathrm{im}}},\\ \frac{1}{R\prime}&=\frac{1}{R}+\frac{1}{r_0}, \end{aligned}
and σim and Eim are the Poisson’s ratio and the Young’s modulus of the impactor.
In this case, the radius of the contact zone a is connected with the relative displacement α by the following relationship:
\displaystyle \begin{aligned} a={R\prime}^{1/2} \alpha ^{1/2}. \end{aligned}
(32)

## Normal Impact of an Elastic Sphere Upon an Elastic Spherical Shell

Let us choose a cylindrical set of coordinates r, θ, z = x3 with the center at the original point of tangency of the sphere and the spherical shell (Fig. 2). Then the equations of motion of the sphere and the contact spot as a rigid whole in the chosen coordinate system have the form
\displaystyle \begin{aligned} m(\widetilde{\dot v_z} +\ddot \alpha )&=- F_{\mathrm{cont}}, \end{aligned}
(33)
\displaystyle \begin{aligned} \rho \pi a^2 h \widetilde{\dot v_z} &=2\pi ah \widetilde{\sigma _{rz}}+F_{\mathrm{cont}}, \end{aligned}
(34)
where Fcont is the contact force and $$\widetilde {\dot v_z}=\dot v_z|{ }_{r=a}$$.
The kinematic condition
\displaystyle \begin{aligned} \widetilde{v_r}=\dot a \end{aligned}
(35)
and the initial conditions
\displaystyle \begin{aligned} \alpha |{}_{t=0}=0, \quad \dot\alpha |{}_{t=0}=V_0, \quad v_z |{}_{t=0}=0, \end{aligned}
(36)
where $$\widetilde {v_r}=v_r|{ }_{r=a}$$, should be added to Eqs. (33) and (34).
Integrating (33) over t and considering the initial conditions (36) yield
\displaystyle \begin{aligned} \widetilde{v_z}=-\dot \alpha -\frac{k}{m}\int_0^{t}\; \alpha ^{3/2} dt +V_0. \end{aligned}
(37)
Eliminating the value $$\widetilde {v_z}$$ from (28) and (37), one of the desired equations is obtained:
\displaystyle \begin{aligned} \widetilde{v_\xi } -\widetilde{v_\lambda }\frac{a}{R}=-\dot \alpha -\frac{k}{m}\int_0^{t} \;\alpha ^{3/2} dt +V_0. \end{aligned}
(38)
To obtain the second desired equation, it is necessary first to eliminate the value $$\widetilde {\dot v_z}$$ from (33) and (34) and then to exclude the value $$\widetilde {\sigma _{rz}}$$ from the equation found at the previous step and from (30) at a time. As a result, it is obtained:
\displaystyle \begin{aligned} \begin{array}{rcl} {} \rho G_1\widetilde{v_\lambda }\;\frac{a}{R}-\rho G_2\widetilde{v_\xi } =-\frac 12\;\rho a \left(\ddot\alpha +\frac{k}{m}\; \alpha ^{3/2} \right) \\ -\frac{k}{2\pi h\sqrt{R\prime} } \;\alpha.\qquad \end{array} \end{aligned}
(39)
Solving the set of equations (38) and (39) with respect to the values $$\widetilde {v_\lambda }$$ and $$\widetilde {v_\xi }$$ yields
\displaystyle \begin{aligned} \begin{array}{rcl} {} \widetilde{v_\xi }R^{-1}\sqrt{R\prime} \alpha=- \frac{1}{\rho (G_1-G_2)} \left[ \frac{k}{2\pi h R } \;\alpha^2 \right. \\ + \frac{\rho R\prime}{2R}\;\alpha ^{3/2}\left(\ddot\alpha +\frac{k}{m} \;\alpha ^{3/2} \right) \\ \left. +\rho G_1\;\frac{R\prime}{R}\;\alpha \left( \dot \alpha +\frac{k}{m}\int_0^{t} \alpha ^{3/2} dt -V_0\right)\right],\quad \end{array} \end{aligned}
(40)
\displaystyle \begin{aligned} \begin{array}{rcl} {} \widetilde{v_\lambda } \alpha^{1/2}=- \frac{1}{\rho (G_1-G_2)} \left[ \frac{k R}{2\pi h R\prime } \right. \\ + \frac 12\;\rho R\;\alpha ^{1/2}\left(\ddot\alpha +\frac{k}{m} \;\alpha ^{3/2} \right) \;\alpha \\ \left.+\rho G_2\;\frac{R}{\sqrt{R\prime}}\left( \dot \alpha +\frac{k}{m}\int_0^{t}\; \alpha ^{3/2} dt -V_0\right)\right].\quad \end{array} \end{aligned}
(41)
Substituting (40) and (41) in relationship (29), which is preliminary multiplied by α1∕2, and considering formula (35), the following governing nonlinear integrodifferential equation with respect to the value α is obtained:
\displaystyle \begin{aligned} &\left[ \frac 12\;\rho \;\alpha ^{1/2}\left(\ddot\alpha +\frac{k}{m} \;\alpha ^{3/2} \right)\right. \\ &\quad \left.+\frac{k }{2\pi h R\prime }\;\alpha \right] \left( \frac{R\prime}{R}\;\alpha +R \right) \\ &\quad +\frac{\rho }{\sqrt{R\prime}}\left( G_1\;\frac{R\prime}{R}\;\alpha +G_2 R\right) \\ &\quad \times \left( \dot \alpha +\frac{k}{m}\int_0^{t} \alpha ^{3/2} dt \right) \\ &\quad +\frac 12\;\rho (G_1-G_2)\sqrt{R\prime}\dot\alpha \\ &\quad =\frac{\rho V_0}{\sqrt{R\prime}}\left( G_1\;\frac{R\prime}{R}\;\alpha +G_2 R\right). \end{aligned}
(42)
In the limiting case, when the radius of the spherical shell tends to infinity R →, Eq. (42) could be reduced to the following:
\displaystyle \begin{aligned} & \frac 12\;\rho \;\alpha ^{1/2}\left(\ddot\alpha +\frac{k}{m} \;\alpha ^{3/2} \right)\\ &\quad +\frac{k }{2\pi h r_0}\;\alpha \\ &\quad +\frac{\rho G_2}{\sqrt{r_0}}\left( \dot \alpha +\frac{k}{m}\int_0^{t} \alpha ^{3/2} dt \right) \\ &\quad +\frac 12\;\rho (G_1-G_2)\sqrt{r_0}\dot\alpha \\ &\quad =\frac{\rho V_0G_2}{\sqrt{r_0}}. \end{aligned}
(43)
A solution of Eq. (42) could be found in the form of the following series with respect to time t:
\displaystyle \begin{aligned} \alpha =V_0t+ \sum_{i=1}^\infty a_i t^{(2i+1)/2}+\sum_{j=2}^\infty b_j t^j, \end{aligned}
(44)
where ai and bj are coefficients to be determined. Fig. 5 Dimensionless time dependence of the dimensionless contact force occurring in the spherical shell impacted by the falling sphere
Substituting (44) into Eq. (42) and equating the coefficients at integer and fractional powers of t, the set of equations for defining the coefficients ai and bj could be found. For example, the first three of them have the form
\displaystyle \begin{aligned} a_1&=-\frac 43 (G_1-G_2)\frac{V_0^{1/2}{R\prime}^{1/2}}{R} <0, \\ b_2&=\frac{2G_2(G_1{-}G_2)}{R} \left( 1\,{+}\, \frac 13 \;\frac{(G_1{-}G_2)R\prime}{G_2R}\right){>}0, \\ a_2&= -\frac{4}{15}\;\frac{kV_0^{1/2}}{\rho \pi hR\prime}-\frac{4}{15}\;V_0^{1/2}{R\prime}^{1/2} \\ &\quad \times \left[ \frac{G_2b_2}{2V_0R\prime}\left(8+\frac 13\;\frac{(G_1-G_2)R\prime}{G_2R} \right)\right.\\ &\quad \left.-\frac{(G_1-G_2)V_0R\prime}{R^3}\right]. \end{aligned}
Thus, the approximate four-term solution of (42) takes the form
\displaystyle \begin{aligned} \alpha =V_0t+ a_1 t^{3/2}+b_2 t^2+a_2t^{5/2}. \end{aligned}
(45)
When R →, the solution for Eq. (47) is reduced to
\displaystyle \begin{aligned} \alpha =V_0t -\frac{4}{15}\;\frac{kV_0^{1/2}}{\rho \pi h r_0}\;t^{5/2}. \end{aligned}
(46)

Substituting the found function α (45), or (46) in the limiting case, in Eq. (31), the final expression for the contact force could be obtained.

The dimensionless time t = tV 0h−1 dependence of the dimensionless contact force $$F^*_{\mathrm {cont}}=F_{\mathrm {cont}}(Eh^2)^{-1}$$ calculated according to (31) and (45) (relationship (62) is utilized in the limiting case) is presented in Fig. 5 for the following ratios of $$\widetilde r=R_{\mathrm {im}}/R$$: 0 (what corresponds to the case of an elastic plate), 0.001, and 0.01. Reference to Fig. 5 shows that the increase in the radius of the shell results in the increase of both the contact duration and the maximum of the contact force.

## Normal Impact of an Elastic Long Hemisphere-Nose Bar Against an Elastic Spherical Shell

At the moment of impact of a bar against a spherical shell (Fig. 2), the shock waves are generated not only in the shell but in the bar (a longitudinal shock wave) as well. This wave propagates along the bar with the velocity $$G_0=\sqrt {E_{\mathrm {im}}\rho _0^{-1} }$$, where Eim and ρ0 are the elastic modulus and density of the bar. Behind the front of this wave, the relationships for the stress σ and velocity v could be obtained using the ray series (Rossikhin and Shitikova, 1995a, 2007a)
\displaystyle \begin{aligned} \sigma ^-=- \sum_{k=0}^\infty \frac{1}{k!}\;\left[ \sigma _{,(k)}\right]\left( t-\frac{x_3}{G_0}\right)^k, \end{aligned}
(47)
\displaystyle \begin{aligned} v ^-=V_0- \sum_{k=0}^\infty \frac{1}{k!}\;\left[ v_{,(k)}\right]\left( t-\frac{x_3}{G_0}\right)^k. \end{aligned}
(48)

It is assumed that the impactor is long enough and reflected waves do not have time to return at the place of contact before the moment of the rebound of the bar from the shell.

Considering that the discontinuities in the elastic bar remain constant during the process of the wave propagation and using the condition of compatibility (Rossikhin and Shitikova, 1995a)
\displaystyle \begin{aligned} G_0\left[ \frac{\partial Z_{,(k-1)}}{\partial x_3}\right]= -[Z_{,(k)}]+ \frac{\delta [Z_{,(k-1)}]}{\delta t}, \end{aligned}
where Z is the function to be found and δδt is the Thomas-derivative (Thomas, 1961), yield
\displaystyle \begin{aligned} \left[ \frac{\partial \sigma _{,(k-1)}}{\partial x_3}\right]= -G_0^{-1} [\sigma _{,(k)}]. \end{aligned}
(49)
With due account for (49), the equation of motion on the wave surface is written in the form
\displaystyle \begin{aligned}{}[\sigma _{,(k)}]=-\rho _0 G_0 [v _{,(k)}]. \end{aligned}
(50)
Substituting (50) in (47) yields
\displaystyle \begin{aligned} \sigma ^-= \rho _0 G_0 \sum_{k=0}^\infty \frac{1}{k!}\;\left[ v_{,(k)}\right]\left( t-\frac{x_3}{G_0}\right)^k. \end{aligned}
(51)
Comparison of relationships (51) and (48) results in
\displaystyle \begin{aligned} \sigma ^-= \rho _0 G_0 (V_0-v^-). \end{aligned}
(52)
At x3 = 0, expression (52) takes the form
\displaystyle \begin{aligned} \sigma_{\mathrm{cont}}= \rho _0 G_0 (V_0-\widetilde v_z-\dot\alpha ), \end{aligned}
(53)
where $$\sigma _{\mathrm {cont}}=\sigma ^-|{ }_{x_3=0}$$ is the contact stress, $$\widetilde v_z+\dot \alpha =v^-|{ }_{x_3=0}$$ is the normal velocity of the displacements of the spherical shell’s points at the place of contact of the bar with the shell, α is the value characterizing the local indentation of the impactor into the shell, and an overdot denotes the time derivative.
Using formula (53), it is possible to find the contact force
\displaystyle \begin{aligned} F_{\mathrm{cont}}= \rho _0 G_0 (V_0-\widetilde v_z-\dot\alpha )\pi a^2. \end{aligned}
(54)
However, the contact force can be determined not only by formula (54) but according to the Hertz’s law as well (31). Therefore
\displaystyle \begin{aligned} \pi a^2\rho _0 G_0 (V_0-\widetilde v_z-\dot\alpha )=k \alpha ^{3/2}, \end{aligned}
whence it follows that
\displaystyle \begin{aligned} \widetilde v_z=-\dot\alpha -\frac{k}{\pi \rho _0 G_0 R\prime}\;\alpha ^{1/2}+V_0. \end{aligned}
(55)
Eliminating the value $$\widetilde v_z$$ from (28) and (55), one of the desired equations is obtained:
\displaystyle \begin{aligned} -\widetilde v_\lambda \;\frac{a}{R} +\widetilde v_\xi =-\dot\alpha -\frac{k}{\pi \rho _0 G_0 R\prime}\;\alpha ^{1/2}+V_0. \end{aligned}
(56)
The second desired equation could be determined by eliminating $$\widetilde {\dot v_z}$$ and $$\widetilde \sigma _{rz}$$ from (34) by virtue of (30) and (55)
\displaystyle \begin{aligned} &\rho G_1\widetilde v_\lambda \;\frac{a}{R} -\rho G_2\widetilde v_\xi\\ &\quad = \frac 12\;\rho a\left(-\ddot\alpha - \frac{k}{2\pi \rho _0 G_0 R\prime}\;\alpha ^{-1/2}\dot\alpha \right) \\ &\quad -\frac{k}{2\pi h \sqrt{R\prime}}\;\alpha.\qquad \end{aligned}
(57)
Solving the set of Eqs. (56) and (57) with respect to the values $$\widetilde v_\lambda$$ and $$\widetilde v_\xi$$ yields
\displaystyle \begin{aligned} &\widetilde v_\xi R^{-1}\sqrt{R^{\prime}}\;\alpha=- \frac{1}{\rho (G_1-G_2)} \left[ \frac 12\; \frac{\rho R^{\prime}}{R}\;\alpha ^{3/2}\right. \\ &\quad \times \left.\Big(\ddot\alpha + \frac{k}{2\pi \rho_0 G_0 R^{\prime}}\;\alpha ^{-1/2}\dot\alpha \right) +\frac{k}{2\pi h \sqrt{R^{\prime}}}\;\alpha^2 \\ &\quad \left. +\rho G_1 \;\frac{R^{\prime}}{R}\;\alpha \left(\dot\alpha +\frac{k}{\pi \rho_0 G_0 R^{\prime}}\;\alpha ^{1/2}-V_0\right)\right],\quad \end{aligned}
(58)
\displaystyle \begin{aligned} &\widetilde v_\lambda \alpha^{1/2}=- \frac{1}{\rho (G_1-G_2)} \left[ \frac 12\; \rho R\;\alpha ^{1/2} \Big(\ddot\alpha \right. \\ &\quad \left.+ \frac{k}{2\pi \rho _0 G_0 R^{\prime}}\;\alpha ^{-1/2}\dot\alpha \right) +\frac{kR}{2\pi h \sqrt{R^{\prime}}}\;\alpha \\ &\quad \left. +\rho G_2 \;\frac{R}{\sqrt{R^{\prime}}} \left(\dot\alpha +\frac{k}{\pi \rho _0 G_0 R^{\prime}}\;\alpha ^{1/2}-V_0\right)\right].\quad \end{aligned}
(59)
Substituting (58) and (59) in (29), which is preliminarily multiplied by α1∕2, the governing nonlinear differential equation with respect to the value α is obtained as
\displaystyle \begin{aligned} &\left[ \frac 12\; \rho \left(\alpha ^{1/2}\ddot\alpha + \frac{k}{2\pi \rho _0 G_0 R\prime}\;\dot\alpha \right) +\frac{k}{2\pi h R\prime}\;\alpha \right] \\ &\quad \times \left( \frac{R\prime}{R}\;\alpha +R \right) \\ &\quad +\frac{\rho} {\sqrt{R\prime}}\left( G_1\;\frac{R\prime}{R}\;\alpha +G_2 R\right) \\ &\quad \times \left( \dot \alpha + \frac{k}{2\pi \rho _0 G_0 R\prime}\;\alpha ^{1/2}\right) \\ &\quad +\frac 12\;\rho (G_1-G_2)\sqrt{R\prime}\dot\alpha \\ &\quad =\frac{\rho V_0}{\sqrt{R\prime}} \left( G_1\;\frac{R\prime}{R}\;\alpha +G_2 R\right).\quad \end{aligned}
(60)
In the limiting case, when the radius of the spherical shell tends to infinity R →, Eq. (60) could be reduced to the following:
\displaystyle \begin{aligned} & \frac 12\; \rho \left(\alpha ^{1/2}\ddot\alpha + \frac{k}{2\pi \rho _0 G_0 r_0}\;\dot\alpha \right) +\frac{k}{2\pi h r_0}\;\alpha \\ &+\frac{\rho} {\sqrt{r_0}}\;G_2 \left( \dot \alpha + \frac{k}{2\pi \rho _0 G_0 r_0}\;\alpha ^{1/2}\right) =\frac{\rho V_0}{\sqrt{r_0}}\; G_2.\quad \end{aligned}
(61)
A solution of (60) could be found in the form of the series (44) with respect to time t. Substituting (44) into Eq. (60) and equating the coefficients at integer and fractional powers of t, the set of equations for defining the coefficients ai and bj could be determined. For example, the first four of them have the form
\displaystyle \begin{aligned} a_1&=-\frac 43\left( \frac{k}{2\pi \rho _0 G_0 R\prime}\right.\\ &\quad \left.+(G_1-G_2)\frac{\sqrt{R\prime}}{R}\right) V_0^{1/2}<0,\\ b_2&=\frac 38\;a_1^2V_0^{-1}\\ &\quad +G_2\left( \frac{k}{2\pi \rho _0 G_0 {R\prime}^{3/2}}+\frac{2(G_1-G_2)}{R}\right),\\ a_2&=\frac{8}{15}\left\{\frac{1}{16}\;a_1b_2V_0^{-1}-\frac 38\;\frac{R\prime}{R^2}\;a_1V_0 -\frac{kV_0^{1/2}}{2\pi \rho hR\prime}\right. \\ &\quad -\frac{kV_0^{1/2}}{4\pi \rho_0 G_0}\left( \frac{V_0}{R^2} +\quad \frac{a_1G_2}{V_0^{3/2}{R\prime}^{3/2}}\right) \\ &\quad +\left. \frac{1}{V_0^{1/2}{R\prime}^{1/2}}\left( \frac{R\prime}{R^2}\;G_1V_0^2-2b_2G_2\right) \right\} , \\ b_3&=-\frac{1}{16}a_1a_2V_0^{-1}-\frac{3}{16}\;\frac{R\prime}{R^2}\;a_1^2-\frac 16\;b_2^2V_0^{-1} \\ &\quad -\frac{ka_1}{6\pi \rho hR\prime V_0^{1/2}} -\frac{5kV_0^{1/2}a_1}{24\pi \rho_0 G_0R^2} \\ &\quad -\frac{1}{6V_0^{1/2}{R\prime}^{1/2}}\left( \frac{R\prime}{R^2}\;G_1V_0^2a_1+5a_2G_2\right) \\ &\quad -\frac{k}{6\pi \rho _0 G_0 {R\prime}^{3/2}}\left( \frac{b_2G_2}{2V_0}+\frac{R\prime}{R^2}\;G_1V_0\right). \end{aligned}
Thus, the approximate five-term solution has the form
\displaystyle \begin{aligned} \alpha =V_0t+ a_1 t^{3/2}+b_2 t^2+ a_2 t^{5/2}+b_3 t^3. \end{aligned}
(62)
In the limiting case, the coefficients in the series (44) representing the solution of equation (61) take the form
\displaystyle \begin{aligned} a_1=-\frac 23\; \frac{kV_0^{1/2}}{\pi \rho _0 G_0 r_0}<0, \end{aligned}
\displaystyle \begin{aligned} b_2=\frac 12\;\frac{k}{\pi \rho _0 G_0 r_0} \left(\frac 13\;\frac{k}{\pi \rho _0 G_0 r_0}+\frac{G_2}{ r_0^{1/2}}\right)>0, \end{aligned}
\displaystyle \begin{aligned} a_2=-\frac{4}{15}\;\frac{k}{\pi \rho _0 G_0 r_0V_0^{1/2}} \left[\frac{2G_2^2}{r_0} +\frac{\rho _0V_0G_0}{\rho h} \right. \end{aligned}
\displaystyle \begin{aligned} \left. +\frac 18\;\frac{k}{\pi \rho _0 G_0 r_0}\left(\frac 19\;\frac{k}{\pi \rho _0 G_0 r_0} +\frac{3G_2}{r_0^{1/2}}\right)\right]<0. \end{aligned}

The dimensionless time t = tV 0h−1 dependence of the dimensionless contact force $$F^*_{\mathrm {cont}}=F_{\mathrm {cont}}(Eh^2)^{-1}$$ calculated according to (31) and (62) is presented in Fig. 6 for the following ratios of $$\widetilde r=R_{im}/R$$: 0 (what corresponds to the case of an elastic plate), 0.001, and 0.01.

From Fig. 6 it is seen that the increase in the radius of the shell results in the increase of both the contact duration and the maximum of the contact force, as it has been mentioned above in the case of the dynamic response of the spherical shell impacted by the elastic sphere. However, the comparison of Figs. 5 and 6 shows that the magnitudes of the contact duration and the maximum of the contact force in the case when the spherical shell is impacted by the cylindrical rod are lower than those when the shell is impacted by the sphere. This is due to the fact that the wave phenomenon is neglected in the falling sphere, while the propagation of the transient waves in the falling rod is taken into account. Fig. 6 Dimensionless time dependence of the dimensionless contact force occurring in the spherical shell impacted by the long cylindrical rod

## Conclusion

The problem on normal low-velocity impact of an elastic falling body upon an elastic Timoshenko-type spherical shell has been analyzed using the wave approach. At the moment of impact, shock waves (surfaces of strong discontinuity) – quasi-longitudinal and quasi-transverse waves – are generated in the target, which then propagate along the body during the process of impact. Behind the wave fronts up to the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and one-term or multiple-term ray expansions, while within the contact region, the nonlinear Hertz’s theory is employed.

For the analysis of the processes of shock interactions of the elastic sphere or elastic spherically headed rod with the Timoshenko-type spherical shell, nonlinear integrodifferential or nonlinear differential equations have been, respectively, obtained with respect to the value characterizing the local indentation of the impactor into the target, which have been solved analytically in terms of time series with integer and fractional powers.

Thus, these examples discussed above show the efficiency of the ray method in solving the impact interaction problems.

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