# Ray Expansions in Impact Interaction Problems

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_99-1

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## Synonyms

## 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 *r*_{0} and mass *m* (Fig. 1) or a long cylindrical elastic rod of radius *r*_{0} with a hemispherical nose of the same radius (Fig. 2) move along the *x*_{3}-axis with the velocity V _{0} towards an elastic isotropic spherical shell of the *R* radius (Rossikhin et al., 2011).

The impact occurs at the initial instant of time at *x*_{3} = *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

*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

*u*

^{1}-curves, where

*u*

^{1}is the distance measured along the straight line segment from the

*C*curve, and choose the distance measured along the

*C*curve as

*u*

^{2}(Fig. 3). The

*u*

^{1}-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)

*g*

_{11}= 1,

*g*

_{22}, and

*g*

_{12}= 0 are the covariant components of the metric tensor of the wave surface.

*c*is a certain constant.

*u*

^{2}are defined by the formula (McConnel, 1957)

*u*

^{1}from −

*h*∕2 to

*h*∕2 yield

*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

*du*

^{1}by \(du_*^1\).

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

*u*

_{i}are the displacement vector components; G is the normal velocity of the wave surface; [

*u*

_{i,j}] = [

*∂u*

_{i}∕

*∂x*

_{j}];

*x*

_{j}are the spatial rectangular Cartesian coordinates;

*ξ*=

*u*

^{1}; \(s_1=u_*^1\); [

*u*

_{i,(k)}] = [

*∂*

^{k}

*u*

_{i}∕

*∂t*

^{k}];

*t*is the time;

*v*

_{i}=

*u*

_{i,(1)};

*λ*

_{i},

*τ*

_{i}, and

*ξ*

_{i}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.

*k*= 0 in (10) yields

*λ*and

*μ*are Lame constants, and

*δ*

_{ij}is the Kronecker’s symbol.

*ξ*

_{i}

*ξ*

_{j}and considering equation

*λ*

_{i}

*λ*

_{j}leads to the equation

*σ*are the elastic modulus and the Poisson’s ratio, respectively.

*λ*

_{i}yields

*ρ*is the density of the shell’s material.

*σ*

_{λλ}] from (15) and (17), the velocity of the quasi-longitudinal wave propagating in the spherical shell is defined as

*λ*

_{i}

*ξ*

_{j}and (16) by

*ξ*

_{i}yields

*v*

_{ξ}] = [

*v*

_{i}]

*ξ*

_{i}.

*σ*

_{λξ}] from (20) and (21), the velocity of the quasi-transverse wave is defined as

*u*

_{λ,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, [

*u*

_{λ,λ}] and [

*u*

_{ξ,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

*λ*

_{i}

*λ*

_{j}and express the values [

*v*

_{λ}]

From the comparison of (24) and (25), it is seen that in the right-hand side of (24), the value [*u*_{τ,τ}] = [*u*_{ij}]*τ*_{i}*τ*_{j} 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 *s*_{2}.

### Governing Equations

*γ*as a small value (Fig. 4), and putting \(\cos \gamma {\approx } 1\), \(\sin \gamma \approx \gamma =aR^{-1}\) yield

*a*is the radius of the contact spot and \(\widetilde {\sigma _{rz}}=\sigma _{rz}|{ }_{r=a}\).

_{cont}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

*k*is the contact stiffness coefficient depending on the geometry of colliding bodies, as well as their elastic constants:

*σ*

_{im}and E

_{im}are the Poisson’s ratio and the Young’s modulus of the impactor.

*a*is connected with the relative displacement

*α*by the following relationship:

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

*r*,

*θ*,

*z*=

*x*

_{3}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

_{cont}is the contact force and \(\widetilde {\dot v_z}=\dot v_z|{ }_{r=a}\).

*t*and considering the initial conditions (36) yield

*α*

^{1∕2}, and considering formula (35), the following governing nonlinear integrodifferential equation with respect to the value

*α*is obtained:

*R*→

*∞*, Eq. (42) could be reduced to the following:

*t*:

*a*

_{i}and

*b*

_{j}are coefficients to be determined.

*t*, the set of equations for defining the coefficients

*a*

_{i}and

*b*

_{j}could be found. For example, the first three of them have the form

*R*→

*∞*, the solution for Eq. (47) is reduced to

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*^{∗} = *t*V _{0}*h*^{−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

_{im}and

*ρ*

_{0}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)

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.

*Z*is the function to be found and

*δ*∕

*δt*is the Thomas-derivative (Thomas, 1961), yield

*x*

_{3}= 0, expression (52) takes the form

*α*is the value characterizing the local indentation of the impactor into the shell, and an overdot denotes the time derivative.

*α*

^{1∕2}, the governing nonlinear differential equation with respect to the value

*α*is obtained as

*R*→

*∞*, Eq. (60) could be reduced to the following:

*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

*a*

_{i}and

*b*

_{j}could be determined. For example, the first four of them have the form

The dimensionless time *t*^{∗} = *t*V _{0}*h*^{−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.

## 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.

## Cross-References

## References

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