Transient wave propagation in Cosserat-type shells

Composites Part B 163 (2019) 145–149

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Transient wave propagation in Cosserat-type shells☆ 1

Yury A. Rossikhin , Marina V. Shitikova

∗

T

Research Center on Dynamics of Solids and Structures, Voronezh State Technical University, 20-letija Oktjabrja Street 84, Voronezh, 3940006, Russia

A R T I C LE I N FO

A B S T R A C T

Keywords: Nano-structures Elasticity Microstructures Analytical modelling

The aim of this paper is to find the magnitudes of velocities of transient waves which could be originated and propagate in thin shells made of Cosserate-type material in the form of surfaces of strong and weak discontinuity. Thus, starting from the 3D Cosserat continuum, the velocities of four transient waves propagating in a thin shell with micro-structure have been determined according to a wave theory for thin-walled plates and shells proposed recently by the authors. Using the expansion ray theory and conditions of compatibility for thin-walled structures, it has been found that (1) the velocities depend only on material constants, and (2) only one micropolar modulus α, which governs the asymmetry of the stress tensor, influences the velocity of the quasi-shear wave, while the Láme moduli λ and μ do not affect the velocities of Cosserat waves, which are generated due to micropolar rotations. This results to the fact that the mathematical theory due to Cosserat is not coupled one. The knowledge of the velocities of transient waves in thin shells made of Cosserat-type materials will allow one to solve boundary-value transient dynamic problems resulting in the propagation of surfaces of strong and weak discontinuity.

1. Introduction

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. 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 [9]. Substituting these series into the boundary conditions on internal and external surfaces of a thin body results in differential equations, and 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 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, which could be overcome only by using advanced numerical treatment [10–12]. What is the main purpose to take shear deformations and rotary inertia into account in boundary-value dynamic problems? The matter is fact that thin-walled bodies in different engineering applications are

One hundred years has passed since Stephen Timoshenko in his pioneer works [1–3] generalized the Bernoulli-Euler beam model introducing into consideration two independent functions: the displacement of the center of gravity of beam's cross-section and rotation of its cross-section around the longitudinal central axis. Further Timoshenko approach was extended to plates and shells [4–7]. The structural mechanics community has celebrated this event by a set of papers, among them the survey [8], wherein an interesting reader could find a lot of references in the field. One of the limitations of the Timoshenko theory is that in the governing equations there exists a certain correction factor k (shear coefficient) which should be determined experimentally, and depending on the character of the experiment it can take on different magnitudes. Thus, k 2 = 5/6 is suggested in Refs. [6,7], the values 2/3 and 8/9 can be found in Refs. [3,4]. Mindlin [5] suggested to determine k 2 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 3D equations of the theory of elasticity. Then the magnitude of k 2 ranges from 0.76 to 0.91 with variation of the Poisson's ratio ν between 0 and 0.5. Sometimes the magnitude k 2 = π 2/12 is in

☆

Some of the results had been presented at ICCE-25 on July 16–22, 2017 in Rome, Italy. Corresponding author. E-mail address: [email protected] (M.V. Shitikova). 1 This paper is one of last by late Professor Rossikhin. ∗

https://doi.org/10.1016/j.compositesb.2018.11.037 Received 30 August 2018; Received in revised form 30 October 2018; Accepted 5 November 2018 Available online 10 November 2018 1359-8368/ © 2018 Elsevier Ltd. All rights reserved.

Composites Part B 163 (2019) 145–149

Y.A. Rossikhin, M.V. Shitikova

often subjected to nonstationary transverse loads, resulting in the generation of transient waves of transverse shear which should be taken into consideration [13,14]. Classical equations describing the dynamic behavior of thin bodies exclude the propagation of such waves. However transient waves propagate in the form of wave surfaces of strong or weak discontinuity. That is why it is quite natural for solving such problems to utilize the theory of discontinuities, which is based on the ray expansions and conditions of compatibility considering transverse deformations of thin bodies. In doing so it is necessary to start from the three-dimensional equations describing the behavior of the material which the thin body is made of. Such a novel approach, despite to the Timoshenko-type theories, does not involve new material constants such as the shear coefficient k, which is rather hard to be determined experimentally for many thinwalled structures, shells or thin-walled beams with open or closed profile as examples. This approach has been developed by the authors since 2007, and during ten years an orderly theory describing the dynamic response of such thin bodies as elastic plates and shells [13], elastic beams [15], elastic spatially curved beams of open profile [16], thermoelastic thinwalled beams [17,18] and thin-walled beams of open profile made of Cosserat pseudo-continuum [19,20], has been created. Recently the interest to the analysis of composite structures using Cosserat theory has been renewed [20–25] owing to the appearance of efficient techniques allowing one to reconstruct Cosserat moduli in materials using long waves [26], or to derive Cosserat moduli via homogenization of heterogeneous elastic materials [27–31]. However it should be noted that in the majority of publications in the field, the authors have limited themselves by the consideration of static problems [21,23,24] or harmonic wave propagation [22,25,30] (an interested reader could also find a lot of useful references within the above mentioned articles [19–31]). In the present paper, the theory proposed for the transient dynamic response of thin-walled beams of open profile made of Cosserat-type materials [19,20] is extended to Cosserat-type shells [32].

Fig. 1. Scheme of the location of the wave and the ray on the shell's middle surface with the corresponding basic vectors.

perpendicular to the shell's middle surface. Besides the ray coordinates, let us introduce also the unit vectors coupled with the ray lines: the vector λ {λi} tangent to the ray curve, and the vectors of the main normal ξ {ξi} and binormal τ {τi} to the ray curve (Fig. 1). Writing each equation from (1)-(4) on both sides of the wave surface and taking the difference of the corresponding equation ahead of and behind the wave surface Σ yield

[σij] = λ [uk, k ] δij + μ ([ui, j] + [uj, i]) + α ([ui, j] − [uj, i] − 2 ∈kij [ψk ]), (5)

2. Problem formulation and governing equations

[μij ] = β [ψk, k ] δij + γ ([ψi, j] + [ψj, i]) + ε ([ψi, j] − [ψj, i]),

(6)

Let us first consider a three-dimensional Cosserat continuum, the dynamical behavior of which is described by the following set of equations [33,34]:

[σij, j] = ρ [v˙ i],

(7)

[μij, j ] + ∈ijk [σjk ] = J [ω˙ i],

(8)

σij = λuk, k δij + μ (ui, j + uj, i ) + α (ui, j − uj, i − 2 ∈kij ψk ),

(1)

μij = βψk, k δij + γ (ψi, j + ψj, i ) + ε (ψi, j − ψj, i ),

(2)

σij, j = ρv˙ i,

(3)

μij, j + ∈ijk σjk = Jω˙ i ,

(4)

where [Z ] = Z+ − Z −, signs + and - denote that an arbitrary function Z is calculated ahead of and behind the wave surface Σ , respectively. Let us interpret the wave surface Σ as a layer of small thickness δ, the forward front of which arrives at a certain point M of the shell at the moment t, while its back front attains the same point at the instant t + Δt , where Δt is s small value. Using the Hadamard-Thomas conditions of compatibility [35] within this layer

where σij is the stress tensor, μij is the moment stress tensor, ∈kij are the Levi-Civita tensor components, ui is the displacement vector, vi = u˙ i is the velocity vector, a dot denotes a time-derivative, an index after a coma labels a derivative with respect to the corresponding coordinate, ψi is the angular rotation vector, ωi = ϕ˙ i is the angular velocity vector, ρ is the density, δij is the Kronecker's symbol, J is the moment of inertia, λ, μ, α, β, γ and ε are material constants, and x i (i = 1,2,3) are Cartesian coordinates. Suppose that a transient wave (surface of strong discontinuity) propagates with the velocity G in a shell made of the micropolar material with the Cosserat pseudo-continuum properties. Along with the Cartesian coordinates, let us introduce the ray coordinates s1, s2 , ξ, where s1 is the coordinate directed along the ray, s2 is the coordinate directed along the line L, which is generated as the result of the intersection of the wave front with the shell's middle surface (Fig. 1), and ξ is directed along the normal to the middle surface. It is assumed that a wave-strip of strong discontinuity during its propagation remains to be

σij . j = −G−1σ˙ ij λj +

μij . j = −G−1μ˙ ij λj +

∂σij ∂s1

λj +

∂μij ∂s1

∂σij

λj +

∂s2

τj +

∂μij ∂s2

∂σij ∂ξ

τj +

ξ j,

∂μij ∂ξ

ξ j,

(9)

(10)

we rewrite Eqs. (3) and (4) inside the shock layer in the form

− G−1σ˙ ij λj +

− G−1μ˙ ij λj +

∂σij ∂s1

λj +

∂μij ∂s1

λj +

∂σij ∂s2

τj +

∂μij ∂s2

τj +

∂σij ∂ξ

ξ j = ρv˙ i,

∂μij ∂ξ

ξ j = Jω˙ i .

(11)

(12)

Fixing the coordinates s1, s2 and ξ in the relationships (11) and (12), then integrating these equations over time from t to t + Δt and tending Δt to zero, we are led to the dynamic conditions of compatibility

[σij] λj = −ρG [vi], 146

(13)

Composites Part B 163 (2019) 145–149

Y.A. Rossikhin, M.V. Shitikova

[μij ] λj = −JG [ωi].

multiplied by λi and from expressions (26) and (28) we find

(14)

Now writing the compatibility conditions for the displacements of the front of the shock wave and considering that there are no cracks, i.e. [ui] = [ψi] = 0 , we find

∂ui ξ j ⎤ [ui, j] = −G−1 [vi] λj + ⎡ ⎢ ∂ξ ⎥, ⎦ ⎣

(15)

∂ψi ξ j ⎤ [ψi, j] = −G−1 [ωi] λj + ⎡ ⎢ ∂ξ ⎥. ⎦ ⎣

(16)

(17)

[ωi] = [ωλ ] λi + [ωτ ] τi + [ωξ ] ξi,

(18)

∂ξ

⎤+⎡ ⎦ ⎣

∂uj ξi ∂ξ

⎤⎞ + α ⎛⎡ ⎦⎠ ⎝⎣

∂ui ξ j ∂ξ

⎤−⎡ ⎦ ⎣

∂uj ξi ∂ξ

⎤ ⎞ + λ [Eξ ] δij, ⎦⎠

G2 =

∂ψi ξ j ∂ξ

⎤+⎡ ⎦ ⎣

∂ψj ξi ∂ξ

⎤⎞ + ε ⎛⎡ ⎦⎠ ⎝⎣

∂ψi ξ j ∂ξ

⎤−⎡ ⎦ ⎣

∂ψj ξi ∂ξ

G1 =

(21)

[eξ ] =

β [ωλ ]. G (β + 2γ )

(32)

E . ρ (1 − ν 2)

(33)

(34)

yield

(22)

corresponding to the condition of nonpressing of shell's layers on each other during the wave front propagation. Considering (22) and multiplying (19) and (20) by ξi ξ j yield

λ [ς ], G (λ + 2μ)

[ωλ ] = [ωi] λi ≠ 0.

∂ψ λ ⎡ ∂uj λj ⎤ = ⎡ j j ⎤ = 0 ⎢ ∂ξ ⎥ ⎢ ⎥ ξ ∂ ⎣ ⎦ ⎣ ⎦

In order to find the values (21) characterizing the transverse deformations of the shell, we utilize the following relationships:

[Eξ ] =

4γ (β + γ ) , J (β + 2γ )

Multiplying further relationships (19) and (20) by λj ξi and assuming that

where

[σij] ξi ξ j = [μij ] ξi ξ j = 0,

(31)

where E is the Young's modulus and ν is the Poisson's ratio, then the velocity G1 is defined by the known formula for the longitudinal wave propagating in elastic plates and shells, namely:

(19)

⎤ ⎞ + β [eξ ] δij, ⎦⎠

∂ψ ξ [eξ ] = ⎡ l l ⎤. ⎢ ⎣ ∂ξ ⎥ ⎦

[ς ] = [vi] λi ≠ 0,

4μ (λ + μ) E = , λ + 2μ 1 − ν2

(20)

∂ul ξl ⎤ [Eξ ] = ⎡ , ⎢ ⎣ ∂ξ ⎥ ⎦

4μ (λ + μ) , ρ (λ + 2μ)

We shall name the first wave Σ1 as the quasi-longitudinal, and the second wave Σ2 as the quasi-torsional micropolar wave. Note that for a particular case of an elastic isotropic shell

[μij ] = −G−1β [ωλ ] δij − G−1γ ([ωi] λj + [ωj] λi ) + G−1ε ([ωj] λi − [ωi] λj ) + γ ⎛⎡ ⎝⎣

(30)

while on another wave

[σij] = −G−1λ [ς ] δij − G−1μ ([vi] λj + [vj] λi ) + G−1α ([vj] λi − [vi] λj ) ∂ui ξ j

⎧JG 2 − 4γ (β + γ ) ⎫ [ωλ ] = 0. ⎨ β + 2γ ⎬ ⎩ ⎭

G1 =

let us rewrite relationships (5) and (6) in the form

+ μ ⎛⎡ ⎝⎣

(29)

From Eqs. (29) and (30) it follows that on one wave

The second terms in (15) and (16) are remained in order take the transverse deformation of the shell into account in further treatment. Considering (15) and (16), as well as the expansions of the values in terms of three mutually orthogonal unit vectors λi , τi and ξi

[vi] = [ς ] λi + [θ] τi + [η] ξi,

⎧ρG 2 − 4μ (λ + μ) ⎫ [ς ] = 0, ⎨ λ + 2μ ⎬ ⎭ ⎩

[σij] λj ξi = −G−1 (μ + α )[η],

(35)

[μij ] λj ξi = −G−1 (γ + ε )[ωξ ].

(36)

Let us add to formulas (35) and (36) the expressions

(23)

[σij] λj ξi = −ρG [η],

(37)

[μij ] λj ξi = −JG [ωξ ],

(38)

which are obtained from the dynamic conditions of compatibility (13) and (14) after its multiplications by ξi . Solving together Eqs. (35) and (37), as well as Eqs. (36) and (38), we find

(24)

3. Determination of transient wave velocities

{ρG 2 − (μ + α )}[η] = 0,

(39)

{JG 2 − (γ + ε )}[ωξ ] = 0,

(40)

Now we could proceed to defining the character of the shock waves propagating in the shell and their velocities. If we multiply Eqs. (19) and (20) by λi λj and consider formulas (23) and (24), then as a result we obtain

G3 =

μ+α , ρ

[η] = [vi] ξi ≠ 0,

4μ (λ + μ) [ς ], [σij] λi λj = − G (λ + 2μ)

(25)

G4 =

γ+ε , J

[ωξ ] = [ωi] λi ≠ 0.

4γ (β + γ ) [ωλ ]. [μij ] λi λj = − G (β + 2γ )

(26)

whence we obtain two waves, Σ3 and Σ4 : (41)

(42)

[σij] λj λi = −ρG [ς ],

(27)

If we apply the similar procedure to Eqs. (19) and (20) multiplying them by λj τi and to Eqs. (13) and (14) multiplying them by τi , then as a result we obtain the same two waves Σ3 and Σ4 which propagate with the same velocities G3 and G4 , on which the additional relationships are fulfilled: [θ] ≠ 0 on Σ3 and [ωτ ] ≠ 0 on Σ4 . In other words, on the wave Σ3

[μij ] λj λi = −JG [ωλ ].

(28)

G = G3,

Let us add relationships (13) and (14) multiplied by λi to Eqs. (25) and (26), then we have

[η] = [vi] ξi ≠ 0,

on the wave Σ4

From relationships (25) and (27), as well as from (13) and (14) 147

[θ] = [vi] τi ≠ 0,

(43)

Composites Part B 163 (2019) 145–149

Y.A. Rossikhin, M.V. Shitikova

4. Conclusion

Table 1 Magnitudes of transient wave velocities in the Cosserat 3D continuum and Cosserat-type thin-walled structures. 3D Cosserat continuum

Thin-walled Cosserat shells

Thin-walled Cosserat beams

(Pal'mov, 1964 [36]) (Rossikhin & Shitikova, 2015 [19]) (1) quasi-longitudinal wave

(present theory)

of open profile (Rossikhin & Shitikova, 2015 [19]) (1) quasilongitudinal-flexuralwarping wave

G1 =

λ + 2μ ρ

G1 =

(2) quasi-shear wave

G2 =

μ+α ρ

due to microstructure β + 2γ J

4μ (λ + μ) ρ (λ + 2μ)

=

E′ ρ

(2) quasi-shear wave

G2 =

(3) quasi-torsional wave

G3 =

(1) quasi-longitudinal wave

μ+α ρ

due to microstructure 4γ (β + γ ) J (β + 2γ )

=

e′ J

(4) quasi-flexural wave

(4) quasi-flexural wave

due to microstructure

due to microstructure

G4 =

γ+ε J

G4 =

γ+ε J

E ρ

(2) quasi-shearrotational wave

G2 =

(3) quasi-torsional wave

G3 =

G1 =

Starting from the 3D Cosserat continuum, the velocities of four transient waves propagating in a thin shell with micro-structure have been determined according to a wave theory for thin-walled plates and shells proposed in Ref. [13]. It has been found that (1) the velocities depend only on material constants, and (2) only one micropolar modulus α, which governs the asymmetry of the stress tensor, influences the velocity of the quasishear wave, while the Láme moduli λ and μ do not affect the velocities of Cosserat waves, i.e., the second and fourth waves which are generated due to micropolar rotations. In other words, mathematical theory due to Cosserat is not coupled one. The knowledge of the velocities of transient waves in thin shells made of Cosserat-type material will allow one to solve boundary-value transient dynamic problems resulting in the propagation of surfaces of strong and weak discontinuity.

μ+α ρ

(3) quasi-torsional wave due to microstructure

G3 =

e J

Acknowledgement

(4) quasi-flexural wave due to microstructure

G4 =

The research described in this publication has been supported by the Ministry of Education and Science of the Russian Federation (Project No. 9.994.2017/4.6).

γ+ε J

References

G = G4,

[ωξ ] = [ωi] ξi ≠ 0,

[ωτ ] = [ωi] τi ≠ 0.

(44)

[1] Timoshenko SP. Course of theory of elasticity (in Russian), Part II. Petrograd. A.E. Kollins Publishing House; 1916. [2] Timoshenko SP. On the correction for shear of the differential equation for transverse vibrations of prismatic bar. Phil Mag 1921;41(245):744–6. Ser 6. [3] Timoshenko SP. Vibration problems in engineering. New York: Van Nostrand; 1928. [4] YaS Uflyand. Wave propagation under transverse vibrations of rods and plates. Prikl Mat Mech 1948;12(3):287–300. (in Russian). [5] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 1951;18(1):31–8. [6] Reissner E. On the theory of bending of elastic plates. J Math Phys 1944;23(4):184–91. [7] Ambartsumyan SA. Theory of anisotropic plates. Moscow: Fizmatlit; 1967. (in Russian). [8] Elishakoff I, Kaplunov Ju, Nolde E. Celebrating the centenary of Timoshenko's study of effects of shear deformation and rotary inertia. Appl Mech Rev 2015;67:060802https://doi.org/10.1115/1.4031965. [9] Reddy JN. A refined nonlinear theory of plates with transverse shear deformation. Int J Solid Struct 1984;20:881–96. [10] Tornabene F, Brischetto S, Fantuzzi N, Bacciocchi M. Boundary conditions in 2D numerical and 3D exact models for cylindrical bending analysis of functionally graded structures. Shock Vib 2016:2373862https://doi.org/10.1155/2016/ 2373862. 2016. [11] Tornabene F, Fantuzzi N, Bacciocchi M. On the mechanics of laminated doublycurved shells subjected to point and line loads. Int J Eng Sci 2016;109:115–64. https://doi.org/10.1016/j.ijengsci.2016.09.001. [12] Tornabene F, Fantuzzi N, Bacciocchi M, Reddy JN. A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation. Compos B Eng 2017;126:162–91. https://doi.org/10.1016/j. compositesb.2017.06.012. [13] Rossikhin YA, Shitikova MV. The method of ray expansions for investigating transient wave processes in thin elastic plates and shells. Acta Mech 2007;189:87–121. https://doi.org/10.1007/s00707-006-0412-x. [14] Rossikhin YA, Shitikova MV. Engineering theories of thin-walled beams of open section. Chapter 2 In: dynamic response of pre-stressed spatially curved thin-walled beams of open profile. Springer briefs in applied sciences and technology. Springer, 3–16, doi:10.1007/978-3-642-20969-7_2. [15] Rossikhin YA, Shitikova MV. The method of ray expansions for solving boundaryvalue dynamic problems for spatially curved rods of arbitrary cross-section. Acta Mech 2008;200:213–38. https://doi.org/10.1007/s00707-008-0016-8. [16] Rossikhin YA, Shitikova MV. Dynamic response of pre-stressed spatially curved thin-walled beams of open profile. Springer briefs in applied sciences and technology Springer; 2011. https://doi.org/10.1007/978-3-642-20969-7. [17] Rossikhin YA, Shitikova MV. Dynamic response of spatially curved thermoelastic thin-walled beams of generic open profile subjected to thermal shock. J Therm Stresses 2012;35:205–34. https://doi.org/10.1080/01495739.2012.637825. [18] Rossikhin YA, Shitikova MV. Thermal shock upon thin-walled beams of open profile. Hetnarski R, editor. Encyclopedia of thermal stresses, vol. 9. Springer; 2014. p. 5146–67. https://doi.org/10.1007/978-94-007-2739-7_928. [19] Rossikhin YA, Shitikova MV. Transient wave velocities in pre-stressed thin-walled beams of open profile with Cosserat-type micro-structure. Compos B Eng 2015;83:323–32. https://doi.org/10.1016/j.compositesb.2015.07007. [20] Rossikhin YA, Shitikova MV. A new approach for studying the transient response of

We shall name the wave Σ3 as the quasi-shear wave, while the wave Σ4 as the quasi-flexural micropolar wave. The prefix ‘quasi-’ in the name of the wave points to the fact that the enumerated waves are the waves of the ‘mixed’ type, i.e., along to the main components characterizing the type of the wave other admixed components could be distinct from zero, however, they are at least of the higher order than the main components. For example, if a certain value experiences a discontinuity on the wave, then on the same wave the first time-derivatives in all other values could have discontinuities. From the found four velocities of propagation of transient waves (surfaces of strong discontinuity) it is seen that (1) they depend only on material constants, and (2) only one micropolar modulus α, which governs the asymmetry of the stress tensor, influences the velocity of the quasi-shear wave G3 , while the Láme moduli λ and μ do not affect the velocities of Cosserat waves, i.e., the second G2 and fourth G4 waves which are generated due to micropolar rotations. In other words, mathematical theory due to Cosserat is not coupled one (the analogous situation takes place in thermoelasticity where there exist uncoupled and coupled theories). Now let us compare the found velocities of the transient waves in the Cosserat-type shell with those in a three-dimensional Cosserat continuum [19,36] and in thin-walled beams of open profile made of Cosserat-type materials [19,20], which are presented in Table, wherein γ (3β + 2γ ) e = β+γ is the micropolar longitudinal modulus, which was introduced in Ref. [20] as the analog of the elastic longitudinal modulus μ (3λ + 2μ) E = λ + μ , and e′ is the reduced micropolar modulus introduced in E

the given paper as the analog of the reduced modulus E′ = 1 − ν2 frequently used in the theory of elastic shells. From the comparison of velocities of transient waves propagating in Cosserat-type thin shells and thin-walled spatially curved beams with those in the Cosserat continuum it is seen (Table 1) that velocities G2 and G4 in all cases are the same, while velocities G1 and G3 are different. The same situation takes place for the case of the transient wave velocities in elastic shells and beams, which could be obtained from the corresponding magnitudes presented in Table by vanishing to zero all additional Cosserat constants except Láme constants.

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