Dynamic Response of a Spherical Shell Impacted by ...

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Dynamic Response of a Spherical Shell Impacted by an Elastic Rod with a Rounded End YURY ROSSIKHIN and MARINA SHITIKOVA Voronezh State University of Architecture and Civil Engineering Research Center on Dynamics of Solids and Structures 20-letija Oktjabrja Street 84, 394006 Voronezh RUSSIA [email protected], [email protected] Abstract: The problem on normal low-velocity impact of a long elastic rod upon an elastic spherical shell is considered. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the impactor and the target, which then propagate along these bodies during the process of impact. Behind the wave fronts upto 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. Nonlinear Hertz’s theory is employed within the contact region. For the analysis of the process of shock interaction of the elastic rod with one rounded end with the spherical shell, a nonlinear differential equation has been obtained with respect to the value characterizing the local indentation of the impactor into the target. Key–Words: Hertzs contact law, spherical shell, ray method, wave theory of impact

1

Introduction

response of the shell including the displacement and stress by the finite element method.

The problem on impact of a rigid body against an elastic isotropic spherical shell has repeatedly considered by different authors using disparate models of shock interaction [1-5]. Thus, Hammel [1] modeled the contact force via a spring in series with a viscous element, i.e., with a help of the Maxwell model, in so doing the local bearing of the shell’s material was ignored. Later Senitskii [2] using the same problem formulation as in [1] and taking the local bearing into account studied this problem. Using the approach which is valid for describing the shock interaction of a sphere with a infinitely stretched classical plate [6], as well as Reissner’s approximate theory for transverse vibrations of shallow shells and the quasistatic Hertzian impact theory, Koller and Busenhart [3] reduced the solution of the problem of the impact response of a thin shallow spherical shell to a nonlinear integro-differential equation with respect to the value characterizing the local indentation of the spherical impactor into the shell. This equation was numerically integrated and its main results were experimentally verified. Recently Her and Liao [6] solved the non-linear integro-differential equation derived in [3] by the numerical scheme of Runge-Kutta method to obtain the time history of the contact force at the impact point of the shell. The contact force is then applied on the apex of the shell in order to investigate the dynamic ISSN: 1792-4294

The normal impact of an elastic sphere upon an elastic isotropic spherical shell was also considered in [7]. The elastic features of the impactor were modeled by a linearly elastic spring, while the equations of motion of the spherical shell were adopted from the paper by Biryukov and Kadomtsev [8], who used the membrane theory of shells for describing the shock interaction of the shell with a spherical impactor. But it should be noted that the application of such a theory in problems of impact interaction is impermissible, because the surface forces change abruptly during the transition through the boundary of the contact domain, but one of the necessary conditions for employing this approximate theory of shells is the smooth variation in components of surface loads [9]. Along with an incorrect problem formulation, there is a lot of other mistakes can be found in the cited paper by Loktev and Loktev [7], including the incorrect equation of the motion of the contact domain and faulty condition of compatibilty which is needed for describing the longitudinal shock wave propagation on the basis of the theory of discontinuities. Such fallacies result in the final relationships for the discontinuities in the physical values to be found, where terms of different dimensions are added with each other, i.e., ”kilometers” are added with ”kilograms”. The surprising thing is that this fact has not been understood 453

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by both the authors and the reviewers of this opus [7]. Moreover, it remains a mystery how the authors could carry out further the numerical investigation of such ’fantastic’ relationships and present its graphical interpretation. In the present paper, the problem on normal lowvelocity impact of a long cylindrical elastic rod upon an elastic spherical shell is considered. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the impactor and the target, which then propagate along these bodies during the process of impact. Behind the wave fronts upto the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and oneterm or multiple-term ray expansions [10]. Nonlinear Hertz’s theory is employed within the contact region, resulting in the differential equation with respect to the value characterizing the local indentation of the impactor into the target.

2

Figure 1: Scheme of the propagating wave-strip along the spherical shell surface where g11 = 1, g22 , and g12 = 0 are the covariant components of the metric tensor of the wave surface. The Gaussian curvature for the linear element (1) is defined by formula [11] √ 1 ∂ 2 g22 K = −√ = 0. g22 (∂u1 )2

Problem formulation

Integrating Eq. (3) and considering formula (2) yields √ g22 = 1 + cu1 , (4)

Let a long cylindrical elastic rod with the radius r0 with a rounded end in the form of a semi-sphere with the same radius move along the x3 -axis with the velocity V0 towards an elastic spherical shell of the R radius. The impact occurs at the initial instant of time at x3 = R. At the moment of impact, two shock ”wavesstrips” (surfaces of strong discontinuity) are generated in the shell, which then propagate along the shell during the process of impact.

2.1

where c is a certain constant. It is known that small distances along the coordinate lines u2 are defined by the formula [11] √ ds2 = g22 du2 , or considering (4) ds2 = (1 + cu1 )du2 .

ds2 − du2 = cu1 , du2

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. 1). 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 [11] 1 2

1

2

2 2

ds = (du ) + g22 (u , u )(du ) ,

and integrate the result relationship with respect to u1 from −h/2 to h/2. As a result we obtain Z

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h/2

−h/2

or Z

ds2 − du2 du1 = 0, du2

h/2

−h/2

ds2 du1 = h. du2

(6)

Equation (6) can be written as 1 h

(1)

in so doing g22 (0, u2 ) = 1,

(5)

Let us rewrite formula (5) in the form

Geometry of the wave surface

2

(3)

Z

h/2

−h/2

√

g22 du1 = 1,

i.e., the mean magnitude of the value thickness of the shell is equal to unit.

(2) 454

√

g22 over the

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If the shell’s thickness is small, then it is possible to consider approximately that √ g22 ≈ 1 (7)

+λ[uξ,ξ ]δij + µ

1 2

2 2

ds ≈ (du ) + (du ) ,

[vξ ] = [vi ]λi ,

[uξ,ξ ] =

(9)

[σλλ ] = −

4µ(λ + µ) −1 G [vλ ], λ + 2µ

E G−1 [vλ ], (15) 1 − σ2 where σ is the Poisson’s ratio. Alternatively, multiplying the three-dimensional equation of motion written in terms of discontinuities [σλλ ] = −

[σij ]λj = −ρG[vi ],

(16)

[σλλ ] = −ρG[vλ ],

(17)

by λi , we obtain

where ρ is the density of the shell’s material. Eliminating the value [σλλ ] from (15) and (17), we find the velocity of the quasi-longitudinal wave propagating in the spherical shell s

G1 =

E . ρ(1 − σ 2 )

(18)

(11)

Relationship (15) with due account for (18) takes the form [σλλ ] = −ρG1 [vλ ]. (19)

Writing the Hook’s law for a three-dimensional medium in terms of discontinuities and using the condition of compatibility (11), we find

Multiplying (12) by λi ξj and (16) by ξi , we have

[σij ] = −G−1 λ[vλ ]δij − G−1 µ ([vi ]λj + [vj ]λi ) ISSN: 1792-4294

(13)

or

δ[ui,(k) ] λj δs1

δ(ui ξj ) . δξ

λ [vλ ]. G(λ + 2µ)

[σλλ ] = [σij ]λi λj = −G−1 (λ + 2µ))[vλ ] + λ[uξ,ξ ]. (14) Substituting (13) in (14) yields

δ[ui,(k) ] δui,(k) ξj + τj + , (10) δs2 δξ 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 , s1 = u1∗ , [ui,(k) ] = [∂ k ui /∂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 [ui,j ] = −G−1 [vi ]λj +

Multiplying relationship (12) from right and from left by λi λj , we are led to the equation

Now we write the condition of compatibility on the wave surface of strong discontinuity. Based on the aforesaid and considering (7)-(9), it takes the form

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, we find

The main kinematic and dynamic characteristics of the wave surface

[σξξ ] = [σij ]ξi ξj = 0,

but considering formula (7) it can be rewritten in the form of (8) by substituting du1 by du1∗ .

[ui,j(k) ] = −G−1 [vi,(k) ]λj +

(12)

Multiplying relationship (12) from right and from left by ξi ξj and considering equation

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 we once again are under the conditions of the McConnel theorem, and thus the linear element of this surface takes the form

2.2

,

δuξ δ(ui ξi ) [uξ,ξ ] = = . δξ δξ

(8)

ds2 = (du1∗ )2 + g22 (du2 )2 ,

where

at any point of the wave surface. Since all values for the shell are averaged over its √ thickness, then such an approximation for g22 is not unreasonable. The linear element (1) with due account for (7) can be approximately written as 2

δ(ui ξj ) δ(uj ξi ) + δξ δξ

[σλξ ] = [σij ]λi ξj = −µG−1 [vξ ], 455

(20)

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[σλξ ] = −ρG[vξ ],

(21)

where [vξ ] = [vi ]ξi . Eliminating the value [σλξ ] from (20) and (21), we find the velocity of the quasi-transverse wave r

G2 =

µ . ρ

(22)

Considering (22), relationship (20) takes the form [σλξ ] = −ρG2 [vξ ].

(23)

Note that in the three-dimensional medium only one value, i.e., [uλ,λ ], is nonzero on the quasilongitudinal 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, we multiply (11) from right and from left by λi λj and express the values [vλ ]

Figure 2: Scheme of velocities and stresses in the shell’s element on the boundary of the contact domain valid, which are the first terms of the ray expansions (Fig. 2), i.e., σλλ = −ρG1 vλ , (26)

[vλ ] = −G1 [uλ,λ ],

σλξ = −ρG2 vξ .

and then the obtained expression we substitute in (13). As a result we find the desired linkage [uξ,ξ ] = −

σ [uλ,λ ]. 1−σ

Considering the cone angle of the contact spot 2γ as a small value (Fig. 2), and putting cos γ ≈ 1, sin γ ≈ γ = aR−1 , we obtain

(24)

vfz = veξ − vfλ

However, if we simply consider the strains in a thin body, for example, a plate in the rectangular Cartesian set of coordinates, assuming that σzz =

3.1

(25)

From the comparison of (24) and (25) it is seen that in the right-hand side of (24) the value [uτ,τ ] = [uij ]τi τj 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 .

(30)

Normal impact of an elastic long bar with a rounded end against an elastic spherical shell

At the moment of impact of a bar against a spherical shell, the shock wave is generated not only in the shell but in the bar (a longitudinal shock wave) as well. This waveqpropagates along the bar with the veloc-

ity G0 = Eim ρ−1 0 , 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 [10]

Governing equations

Thus, behind the front of each of two transient waves (surfaces of strong discontinuity) upto the boundary of the contact domain relationships (19) and (23) are ISSN: 1792-4294

(29)

g where a is the radius of the contact spot, and σ rz = σrz |r=a .

then it is possible to obtain a little bit another formula

3

(28)

a + vfλ , R a g f σ − ρG2 veξ , rz = ρG1 v λ R

+σ(ux,x + uy,y )) = 0,

σ (ux,x + uy,y ). 1−σ

a , R

ver = veξ

E ((1 − σ)uz,z (1 + σ)(1 − 2σ)

[uz,z ] = −

(27)

σ− = −

∞ X 1 h k=0

456

k!

σ,(k)

i

t−

x3 G0

k

,

(31)

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v − = V0 −

∞ X 1 h

k! k=0

v,(k)

i

t−

x3 G0

k

.

σim , Eim and Rim are the Poisson’s ratio, Young’s modulus and the radius of the impactor, and σ, E and R are the same values for the target. Thus,

(32)

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

G0

πa2 ρ0 G0 (V0 − vez − α) ˙ = kα3/2 , whence it follows that vez = −α˙ −

where Z is the function to be found, and δ/δt is the Thomas-derivative [12], we have ∂σ,(k−1) = −G−1 0 [σ,(k) ]. ∂x3

−veλ

(33)

With due account for (33) the equation of motion on the wave surface is written in the form

g ρπa2 hvf ˙ z = 2πahσ rz + Fcont ,

(34)

∞ X 1 h k=0

k!

v,(k)

i

t−

x3 G0

k

.

ρG1 veλ

k √ α. (42) 2πh R0 Solving the set of equations (40) and (42) with respect to the values veλ and veξ , we have −

(36)

At x3 = 0, expression (36) takes the form σcont = ρ0 G0 (V0 − vez − α), ˙

(37)

veξ R

σ−|

ez + α˙ = where σcont = x3 =0 is the contact stress, v v − |x3 =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 (37), it is possible to find the contact force 2

Fcont = ρ0 G0 (V0 − vez − α)πa ˙ .

1 − σ2 , E

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k 00 =

−1

√

R0

1 1 ρR0 3/2 α=− α α ¨ ρ(G1 − G2 ) 2 R

k k √ α2 α−1/2 α˙ + + 2πρ0 G0 R0 2πh R0 R0 k 1/2 +ρG1 α α˙ + α − V0 , (43) R πρ0 G0 R0 1 1 veλ α1/2 = − ρR α1/2 α ¨ ρ(G1 − G2 ) 2

k kR √ α + α−1/2 α˙ + 0 2πρ0 G0 R 2πh R0 k R 1/2 +ρG2 √ α˙ + α − V0 . (44) πρ0 G0 R0 R0 Substituting (43) and (44) in (29), which is preliminary multiplied by α1/2 , we obtain the governing nonlinear differential equation with respect to the value α

(38)

However, the contact force can be determined not only by formula (38) but according the Hertz’s law as well Fcont = kα3/2 , √ R0 1 4 1 1 k= , = + , 0 00 0 3π k + k R Rim R k0 =

a 1 k −ρG2 veξ = ρa −¨ α− α−1/2 α˙ R 2 2πρ0 G0 R0

(35)

Comparing relationships (35) and (32), we obtain σ − = ρ0 G0 (V0 − v − ).

(41)

erz where vf ˙ z = v˙ z |r=a , and then eliminate vf ˙ z and σ from (41) by virtue of (30) and (39). As a result we obtain

Substituting (34) in (31) yields σ − = ρ0 G 0

a k α1/2 + V0 . (40) + veξ = −α˙ − R πρ0 G0 R0

The second desired equation we obtain if we first write the equation of motion of the contact spot as a rigid whole in the chosen coordinate system in the following form

[σ,(k) ] = −ρ0 G0 [v,(k) ].

(39)

Eliminating the value vez from (28) and (39), we are led to one of the desired equations

∂Z,(k−1) δ[Z,(k−1) ] = −[Z,(k) ] + , ∂x3 δt

k α1/2 + V0 . πρ0 G0 R0

2 1 − σim , Eim

457

1 k k √ α ρ α1/2 α ¨+ α˙ + 0 2 2πρ0 G0 R 2πh R0

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×

R0 α+R R

R0 ρ k G1 α1/2 +√ α + G2 R α˙ + 0 R 2πρ0 G0 R0 R √ 1 ρV0 R0 0 √ + ρ(G1 − G2 ) R α˙ = G1 α + G2 R . 2 R R0 (45) We will seek a solution of (45) in the form of the following series with respect to time t:

α = V0 t +

∞ X

ai t(2i+1)/2 +

i=1

∞ X

bj tj ,

(46)

j=2

Figure 3: Dimensionless time dependence of the dimensionless contact force

where ai and bj are coefficients to be determined. Substituting (46) into equation (45) and equating the coefficients at integer and fractional powers of t, we are led to the set of equations for defining the coefficients ai and bj . For example, the first two of them have the form √ ! k R0 4 1/2 a1 = − + (G − G ) V0 < 0, 1 2 0 3 2πρ0 G0 R R 3 b2 = a21 V0−1 +G2 8

k

2(G1 − G2 ) + R

References: [1] J. Hammel, Aircraft Impact on a Spherical Shell, Nuclear Eng. Design 37, 1976, pp. 205–223. [2] Yu.E. Senitskii, Impact of a Viscoelastic Solid Along a Shallow Spherical Shell, Mech. Solids (English translation) 17, 1982, pp. 120–124. [3] M.G. Koller and M. Busenhart, Elastic Impact of Spheres on Thin Shallow Spherical Shells, Int. J. Impact Eng. 4, 1986, pp. 11–21. [4] D.I. Lee and B.M. Kwak, An Analysis of LowVelocity Impact of Spheres on Elastic CurvedShell Structures, Int. J. Solids Structures 30, 1993, pp. 2879–2893. [5] S.-C. Her and C.-C. Liao, Analysis of Elastic Impact on Thin Shell Structures, Int. J. Modern Phys. B 22, 2008, pp. 1349–1354. [6] C. Zener, The Intrinsic Inelasticity of Large Plates, Phys. Reviews 59, 1941, pp. 669–673. [7] A.A. Loktev and D.A. Loktev, Transverse Impact of a Ball on a Sphere with Allowance for Waves in the Target, Tech. Phys. Letters (English translation) 34, 2008, pp. 960–963. [8] D.G. Biryukov and I.G. Kadomtsev, Dynamic Elastoplastic Interaction Between an Impactor and a Spherical Shell, J. Appl. Mech. Tech. Phys. (English translation) 43, 2002, pp. 777–781. [9] I.A. Birger and Ya.G. Panovko (Editors), Handbook: Strength, Stability, Vibrations (in Russian), Mashinostroenie, Moscow 1968 [10] Yu.A. Rossikhin and M.V. Shitikova, Transient Response of Thin Bodies Subjected to Impact: Wave Approach, Shock Vibr. Digest 39, 2007, pp. 273–309. [11] A.J. McConnel, Application of Tensor Analysis, Dover Publications, New York 1957 [12] T.Y. Thomas, Plastic Flow and Fracture in Solids, Academic Press, New York 1961

!

2πρ0 G0 R0 3/2 Thus, the approximate three-term solution takes the form α = V0 t + a1 t3/2 + b2 t2 . (47)

The dimensionless time t∗ dependence of the di∗ calculated according mensionless contact force Fcont to (47) is presented in Fig. 3 for the following ratios of Rim /R: 0 (curve 1), 0.001 (curve 2), 0.01 (curve 3). Reference to Fig. 3 shows that the decrease in the radius of the shell results in the decrease of both the contact duration and the maximum of the contact force.

4

Conclusion

For the analysis of the process of shock interaction of an elastic rod with one rounded end with a spherical shell, a nonlinear differential equation has been obtained with respect to the value characterizing the local indentation of the impactor into the target, the analytical solution of which is found in terms of time series with integer and fractional powers. Numerical calculations show that the decrease in the radius of the shell results in the decrease of both the contact duration and the maximum of the contact force. Acknowledgements: The research described in this publication was made possible in part by Grant No. 2.1.2/520 from the Russian Ministry of High Education. ISSN: 1792-4294

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ISBN: 978-960-474-203-5