M X Shi

Q M Li

Instability in the Transformation between Extensional and Flexural Modes in Thin-Walled Cylindrical Shells

Keywords: thin-walled shell, extensional mode, flexural mode, mode transformation, numerical instability

With the inclusion of all terms up to the third-order in the membrane strain to consider the geometric nonlinearity in deformation, a third-order implicit model and a fourth-order explicit model for the vibration transformation between extensional and flexural modes in thin-walled cylindrical shells are established and solved numerically. Numerical instability is observed in numerical solutions based on the explicit model. It is found that such numerical instability does not result from the accumulated numerical errors in the numerical integration process, but from the neglecting of higher order terms in the formulation of the problem. With the inclusion of all terms up to the fourth-order in the strain energy, the explicit model can predict a stable vibration history with periodic mode transformations between the two modes. The same 2:1 internal resonance of the vibration mode transformation is predicted by both models. As flexural stress growth is concerned, the two models matches very well during the first group of peaks but lag in phase gradually appears in later group of stress peaks based on the implicit model. This is understood to be resulting from the limited number of terms included in the series expansions based on the explicit model, which allows the most likely excited flexural mode get a larger share of energy transferred from the principle mode and as a result flexural stress arrives at peaks earlier based on the explicit model.

Introduction

The investigation on nonlinear dynamics of circular cylindrical shells has received considerable attentions since the middle of fifties of last century. The nonlinearity may come from sources such as geometry, material, inertia or any combination of them. The geometric nonlinearity stems either from the initial boundary conditions or nonlinear strain-and/or curvature-displacement relations. The material nonlinearity occurs when the stresses are non-linear functions of strains, or vice versa. The inertia nonlinearity may be caused by the presence of non-uniform mass distribution. The nonlinearity of the problem eventually appears either in the governing partial differential equations and/or in the boundary conditions. Most of the present studies deal with geometric nonlinearities and the main differences among them are the approximations used in establishing strain and/or curvature relations with displacements. A series of reviews on nonlinear dynamics of shell structures in past decades have been published with emphasis on different subjects of the topic [START_REF] Evensen | Non-Linear Vibrations of Circular Cylindrical Shells[END_REF][START_REF] Leissa | Non-Linear Analysis of Plates and Shell Vibrations[END_REF][START_REF] Qatu | Review of Shallow Shell Vibration Research[END_REF], Liew et al., 1997, Kubenko and Koval'chuk, 1998, [START_REF] Moussaoui | Non-Linear Vibrations of Shell-Type Structures: A Review with Bibliography[END_REF][START_REF] Amabili | Review of Studies on Geometrically Nonlinear Vibrations and Dynamics of Circular Cylindrical Shells and Panels, with and without Fluid-Structure Interaction[END_REF][START_REF] Krysko | Stability, Bifurcation and Chaos of Closed Flexible Cylindrical Shells[END_REF]. Besides, effects of transverse deformation and rotary inertia for moderately thick shells were reviewed in reference [START_REF] Liew | Vibration of Shallow Shells: A Review with Bibliography[END_REF]). An extensive review with 356 references listed in chronological order on the subject of dynamics of circular cylindrical shells was given in reference [START_REF] Amabili | Review of Studies on Geometrically Nonlinear Vibrations and Dynamics of Circular Cylindrical Shells and Panels, with and without Fluid-Structure Interaction[END_REF].

The pioneering works in (Grigolyuk, 1955a,1955b, Reissner, 1955a,1955b) established the nonlinear partial differential equations governing vibrations of cylindrical and spherical shells with finite deflections. Mode expansion method in the nonlinear dynamic analysis of rings and cylindrical shells was introduced in [START_REF] Evensen | Some Observations on the Nonlinear Vibration of Thin Cylindrical Shells[END_REF][START_REF] Evensen | Non-Linear Flexural Vibrations of Thin-Walled Circular Rings[END_REF][START_REF] Evensen | Nonlinear Flexural Vibrations of Thin-Walled Circular Cylinders[END_REF], which strongly supported earlier experimental observations of nonlinear effects of large amplitude on vibration frequencies in finite length circular cylindrical shells. Some classic theoretical and experimental developments on nonlinear dynamics of circular rings and cylindrical shells were also published in [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF][START_REF] Evensen | Nonlinear Flexural Vibrations of Thin Circular Rings[END_REF][START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF][START_REF] Dowell | On Nonlinear Flexural Vibrations of Rings[END_REF][START_REF] Dowell | Modal Equations for the Nonlinear Flexural Vibrations of a Cylindrical Shell[END_REF][START_REF] Mcivor | Dynamic Response of Finite-Length Cylindrical Shells to Nearly Uniform Radical Impulse[END_REF][START_REF] Evensen | Non-Linear Vibrations of Circular Cylindrical Shells[END_REF][START_REF] Lindberg | Stress Amplification in a Ring Caused by Dynamic Instability[END_REF]. On one hand, the well-known Donnell shallow shell theory [START_REF] Donnell | A Discussion of Thin Shell Theory[END_REF] were employed to study the amplitude-frequency characteristics (AFCs) of finite length cylindrical shells [START_REF] Evensen | Nonlinear Flexural Vibrations of Thin Circular Rings[END_REF][START_REF] Dowell | On Nonlinear Flexural Vibrations of Rings[END_REF][START_REF] Evensen | Nonlinear Flexural Vibrations of Thin-Walled Circular Cylinders[END_REF][START_REF] Dowell | Modal Equations for the Nonlinear Flexural Vibrations of a Cylindrical Shell[END_REF]), which has continuously attracted attentions from scientists [START_REF] Dowell | Comments on the Nonlinear Vibrations of Cylindrical Shells[END_REF][START_REF] Amabili | Further Comments on Nonlinear Vibrations of Shells[END_REF][START_REF] Evensen | Nonlinear Vibrations of Cylindrical Shells -Logical Rationale[END_REF]. On the other hand, in McIvor and his colleagues' publications on rings and circular cylindrical shells [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF][START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF][START_REF] Lovell | Nonlinear Response of a Cylindrical Shell to an Impulsive Pressure[END_REF], vibration features of rings and circular cylindrical shells under nearly uniform radial impulse were characterized by a parametric excitation of axisymmetric and/or asymmetric modes. By means of Lagrange equation, they reduced the ordinary-differential equations to a series of nonlinear equations revealing the coupling between the radial breathing mode and flexural modes. Further linearization of the coupled non-linear differential equations led to a Mathieu-type equation, which indicates the stable and unstable regions of the problem and demonstrates the 2:1 internal resonance mechanism when the most likely mode transformations can be initiated from the first unstable region of Mathieu stability diagram [START_REF] Mclachilan | Theory and Application of Mathieu Functions[END_REF][START_REF] Struble | Nonlinear Differential Equations[END_REF]. Intermittent mode transformations could be numerically predicted for circular cylindrical shells under certain circumstances and the earliest study on this topic should be credited to Goodier and his colleagues for their pioneering work [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]McIvor, 1964, McIvor, 1966). Later Nayfeh and Raouf used multiple scale method to get a perturbation solution of the same problem [START_REF] Nayfeh | Non-Linear Oscillation of Circular Cylindrical-Shells[END_REF]. The perturbation approach is effective in delineating mode interactions when frequencies of several successive modes are closely grouped together. In their work, the phenomenon of energy saturation of the driven mode (radial breathing mode) was observed and it was also predicted that the mode transformations might not be steady-state periodic. Instead, cylindrical shells might experience amplitude-and phase-modulated chaotic vibrations. In the case of simply supported circular cylindrical shells filled with a quiescent fluid, it is shown that the coupling between driven mode and excited modes is always linked nonlinearly in the normal form of the system and an internal resonance 1:1:1:2 phenomenon was observed by Amabili and his colleagues using an analytic approach, i.e. the method of normal mode forms [START_REF] Amabili | Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 1: Equations of Motion and Numerical Results[END_REF]. This problem was further investigated by a perturbation analysis [START_REF] Pellicano | Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells. Part 2: Perturbation Analysis[END_REF] focusing on mode interactions and the same nonlinearly coupled vibrations and internal resonance phenomena were found. For thin-walled spherical shell structures, extensive studies have also been carried out with focus analogously on the intrinsic nonlinearities and internal resonance characterizing vibrations [START_REF] Mcivor | Axisymmetric Response of a Closed Spherical Shell to a Nearly Uniform Radial Impulse[END_REF][START_REF] Ye | The Non-Linear Vibration and Dynamic Instability of Thin Shallow Shells[END_REF][START_REF] Nayfeh | Nonlinear Interactions[END_REF][START_REF] Thomas | Non-Linear Vibrations of Free-Edge Thin Spherical Shells: Modal Interaction Rules and 1 : 1 : 2 Internal Resonance[END_REF][START_REF] Nayfeh | Axisymmetric Vibrations of Closed Spherical Shells: Equations of Motion and Bifurcation Analysis[END_REF][START_REF] Touze | Non-Linear Behaviour of Free-Edge Shallow Spherical Shells: Effect of the Geometry[END_REF][START_REF] Thomas | Non-Linear Vibration of Free-Edge Thin Spherical Shells: Experiments on a 1:1:2 Internal Resonance[END_REF][START_REF] Camier | Non-Linear Vibrations of Imperfect Free-Edge Circular Plates and Shells[END_REF].

Apart from the previously mentioned classical shell theory, i.e. the Donnell shell theory or the similar theory under other names [START_REF] Thomas | Non-Linear Vibrations of Free-Edge Thin Spherical Shells: Modal Interaction Rules and 1 : 1 : 2 Internal Resonance[END_REF], more refined thin shell theories have been proposed, such as ⑴ Sanders-Koiter theory [START_REF] Sanders | An Improved First Approximation Theory for Thin Shells[END_REF][START_REF] Yamaki | Elastic Stability of Circular Cylindrical Shells[END_REF], ⑵ Fl ü gge-Lur'ye-Byrne theory [START_REF] Flugge | Statik Und Dynamic Der Schalen[END_REF][START_REF] Lur'ye | General Theory of Elastic Shells[END_REF][START_REF] Byrne | Theory of Small Deformation of a Thin Elastic Shell[END_REF], and ⑶ Novozhilov's theory [START_REF] Novozhilov | The Theory of Thin Elastic Shells[END_REF]. Among these thin shell theories, one common feature is that they all employed kinematic and mechanics hypotheses to describe the three-dimensional deformation of a representative shell element by deformation of the middle surface as a two-dimensional mechanics problem. Differences among them are the various assumptions made on the form of small terms included in strain and curvature expressions and the order of small terms retained in strain-and/or curvature-displacement relations. It may be necessary to point out that only linear terms are included in the curvature expressions in Donnell and Sander-Koiter's theories. Above four shell theories and an improved Novozhilov theory were evaluated in comparison with each other in a study of nonlinearity characteristics of thin circular cylindrical shells with large amplitude deformation [START_REF] Amabili | A Comparison of Shell Theories for Large-Amplitude Vibrations of Circular Cylindrical Shells: Lagrangian Approach[END_REF]. For moderately thick circular cylindrical shells, a first-order shear deformation theory was established by Reddy and his colleagues (Reddy, 1984a,1984b, Reddy and Chandrashekhara, 1985a,1985b[START_REF] Reddy | A Higher-Order Shear Deformation-Theory of Laminated Elastic Shells[END_REF], which is actually a thick-shell version of the Sanders-Koiter thin shell theory where five independent variables (including three displacements and two rotations) are used to describe the deformation.

Only linear terms of the five deformation variables are included in the middle surface curvatures in this first-order moderately thick shell theory. The circumferential curvature is the main concern in the study of flexural deformation of circular cylindrical shells. In contrast to the assumed forms of small terms in previously mentioned four thin shell theories and completely linear terms in Reddy and his colleagues' thick shell theory, the derivation of curvature expression in Goodier and McIvor's theory is mathematically more rigorous [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]McIvor, 1964, McIvor, 1966).

The radial "breathing" mode (RBM) in a cylindrical shell may be initiated by various means, e.g. when it is subjected to a nearly uniform radial impulse. Under certain conditions, strong flexural motion may be developed from unavoidable existence of imperfections in the radial impulse and/or geometric imprecations. Interesting intermittent transformations between RBM and flexural mode(s) may occur subsequently, accompanied with energy transfer between vibration modes [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF][START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF][START_REF] Mcivor | Dynamic Response of Finite-Length Cylindrical Shells to Nearly Uniform Radical Impulse[END_REF]). Recently, it has been revealed that vibration mode transformation is a common phenomenon that may happen in engineering structures at very different length scales governed by 2:1 or similar internal resonance mechanism. For example, strain growth in explosion containment vessels is related to the vibration mode transformation [START_REF] Zhu | Dynamic Response of Cylindrical Explosive Chambers to Internal Blast Loading Produced by a Concentrated Charge[END_REF][START_REF] Duffey | Strain Growth in Spherical Explosive Chambers Subjected to Internal Blast Loading[END_REF], Li et al., 2008) and the intermittent mode transformation in single-walled carbon nanotubes was predicted in [START_REF] Li | Intermittent Transformation between Radial Breathing and Flexural Vibration Modes in a Single-Walled Carbon Nanotube[END_REF], Shi et al., 2009a,2009b, Shi et al., 2009c). Since many natural, bio-and engineering structures have similar geometries to cylindrical and spherical shells, the vibration mode transformation may be realised in all these structures. However, in our recent publications to study the intermittent transformation between RBM and flexural vibration modes in a single-walled carbon nanotube [START_REF] Li | Intermittent Transformation between Radial Breathing and Flexural Vibration Modes in a Single-Walled Carbon Nanotube[END_REF], it was found that some discarded higher order terms in the formulation of problem [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF][START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF][START_REF] Mcivor | Dynamic Response of Finite-Length Cylindrical Shells to Nearly Uniform Radical Impulse[END_REF]) may need to be retained in the strain energy expression to ensure stable vibration mode transformations, which will be studied in details in this paper with focusing on an infinitely long circular cylindrical shell, or thin-walled circular continuum ring, the same classic problem studied decades ago [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]McIvor, 1964, McIvor, 1966).

We will include all terms up to the fourth-order in the strain energy in this study such that the conservation of the total energy in the insulated mechanics system can be guaranteed. The basic formulation of the problem in terms of two non-dimensional variables is given in the first half part of section 2, which is called an implicit model for the problem. In the second half part of section 2 an explicit model is introduced by expanding the two variables into series and then using Lagrange equations to obtain the governing equations of the unknown coefficient functions of the series. Both models are a natural extension of the early formulation in [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]. Section 3 presents the numerical results based on explicit model, focusing on the stability of vibration mode transformations and its dependence on the fourth-order term in strain energy. In section 4 the numerical results based on implicit model are given, focusing on flexural stress growth. The differences between explicit and implicit models are also presented in this section through one numerical example. Finally, main findings of this improved formulation of problem are summarized.

Formulation of the problem

Implicit model for mode transformation

When an infinitely long cylindrical shell undergoes deformation independent of axial coordinate, the original problem can be reduced to a two-dimensional plane strain problem. As shown in ( ) ( ) ( )

1 1 cos sin r p Q a S h a a θ ε κ ε ρ β β ′ + + - + = - + , (2.1)
( ) ( )

1 sin cos r S a Q h a a θ κ ε ρ β β ′ + + = - + , (2.2)
where the prime here and all thereafter indicates partial derivatives with respect to angle θ . In Eqns.(2.1,2), p is the pressure imposed on shell surface, ρ the mass density, S the normal force (compression positive), and Q the shear force, respectively. ε and κ are the membrane strain and curvature at the point ( ) , a θ , respectively, and will be defined later. β is the angle between surface normal and the radius vector r r , which, in a sense of infinitesimal deformation, (since dr and rdφ are the displacements, respectively, in the radial and tangent directions), can be determined by ( ) ( )

tan dr rd r r β φ φ ′ ′ = = . ( 2.3)
When centroid of the cross-section is taken as fixed, acceleration components r a and a θ of the element in Eqns.(2.1,2) are given as 2 , 2 r a r r a r r

θ φ φ φ = - = + & && & && & , (2.4a,b)
where the dot above symbols refers to derivatives with respect to time t . When rotary inertia of the element is neglected, the moment equilibrium equation can lead to ( )

0 1 = + + ′ aQ M ε . (2.5)
Considering the relationship between moment and shear force given by Eqn.(2.5), the governing Eqns.(2.1, 2) can be reduced to a set of equations of normal force S and moment M .

For the deformation shown in Fig. 2.2, the membrane strain (engineering) and curvature at the point ( )

, a θ can be determined, respectively, by

1 2 2 2 0 0 1 ds ds a r r ds ε φ - - ′ ′ = = + -, (2.6)
( ) ( ) ( ) 3
φ β κ φ φ φ φ - -   ′ ′ ′′ ′ ′′ ′ ′ ′ = = - + + +   . ( 2.7)
The relationship between angle β and ( ) , r φ given by Eqn.(2.3) is used to derive the curvature in Eqn.(2.7). Under the assumption of small deformation, i.e. the surface normal and thickness of the shell wall remain unchanged during deformation, the shear deformation along thickness is ignored. Then, for a point at distance z above middle surface, as shown in Fig. 2.3, the strain ( )

z θ ε
and stress ( )

z θ σ
for plane strain can be obtained as ( ) ( )( )

1 1 1 z z z a a a θ ε ε ε κ   = + + - -     , (2.8)
( ) ( )

1 z E z θ θ σ ε = , (2.9)
in which the membrane stiffness 1 E and flexural stiffness D are defined as ( )

2 3 2 1 1 12 , 1 ν ν - = - = Eh D E E , (2.10)
where E and ν are Young's modulus and Poisson's ratio, respectively. Then the normal force S and moment M in Eqns.(2.1, 2,5) can be easily obtained based on the stress distribution along thickness given by Eqn.(2.9), i.e.

( ) ( )( )

2 1 2 1 1 , h h S z dz E h D a θ σ ε ε κ - = - = - + + - ∫ (2.11) ( ) ( )( ) 2 2 1 1 h h M z zdz D a a θ σ ε κ - = = + - ∫ . (2.12)
In order to formulate the problem in a non-dimensional base, two non-dimensional deformation variables ξ (radial displacement, inward positive) and ψ (circumferential rotation, counterclockwise positive) are introduced, i.e.

θ φ ψ ξ - = - = , 1 a r . (2.13a,b)
Then, the membrane strain ε and curvature κ in Eqns.(2.6, 7) can be expressed as ( )

2 2 1 1 2 2 ε ψ ξ ξψ ξ ξ ψ ξ ′ ′ ′ ′ ′ = -- + - -, (2.14)
( ) ( )

2 2 2 1 2 1 ξ ξ ξ ψ ξ ψ ξ ξ ξ κ ′ + + - ′ ′ ′ - ′ ′ ′ - + ′ ′ + = a . (2.15)
It should be pointed out that the membrane strain given by Eqn. (2.14) is different from the one in the early publication (Eqn.11 in [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]). In the present membrane strain expression in Eqn. (2.14), additional third-order terms underlined in the expression are included in the higher order governing equations (up to third order) to avoid the numerical instability during the vibration mode transformation discussed in the introduction section. However, third-order terms are discarded completely in the curvature expression since they do not contribute to the same accuracy level in the final governing equations as their counterparts in membrane strain do.

After eliminating shear force Q using Eqn.(2.5) and substituting accelerations using Eqn.(2.4a,b), with Eqns. (2.11-13) taken into account, the governing Eqns.(2.1,2) are rewritten using the two non-dimensional variables ξ and ψ as , c is the propagation velocity of longitudinal waves and defined as

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) , 0 1 2 3 2 1 2 1 2 1 1 1 ) 4 ( 2 2 2 2 2 2 = ′ - - ′ + - Π + + ′ ′ + + ′ ′ ′ ′ + ′ - ′ ′ + ′ ′ - ′ ′ + ′ ′ + ′ + ′ + ′ ′ ′ - ′ + ′ - ′ - ′ ′ + + - + ξ ψ ξ ψ ξ ξ ξ α ξ ψ ψ ψ ξ ξ ξ ξ ψ ξ ξ ξ ψ ξ ψ ξ ξ ψ ψ ψ ξ ξ ψ ξ ξ p h E a & & & (2.16) ( ) ( ) ( ) ( )( ) ( ) [ ] ( ) , 0 1 2 3 2 2 1 2 1 2 1 ) 4 ( 2 2 2 = ′ - ′ + + + ′ ′ + ′ + ′ ′ ′ + ′ ′ ′ + - - ′ ′ ′ ′ + ′ ′ ′ + ′ ′ + ′ ′ - ′ ′ + ′ ′ - ′ + + - ξ ξ ψ ξ ξ ξ ξ ξ ξ ξ ξ α ξ ψ ξ ξ ψ ξ ψ ξ ξ ψ ξ ψ ξ ψ ξ ξ ψ p h E
ρ 1 E c = . The aspect ratio α is a h 3 2 = α . ( 2.18)
It should be pointed out that all the underlined terms in Eqns. (2.16,17) are additional higher order terms compared with the governing equations in Goodier and McIvor's work [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]McIvor, 1964, McIvor, 1966) and most of them are originated from the inclusion of the third order term in membrane strain given in Eqn. (2.14). The term Π in Eqn.(2.16) is

( ) ( ) ( ) ( ) ( ) (4)
(4) 2 2 3 2 4 2 . 2 ψ ξ ψ ξ ψ ξ ξ ψ ξ ξ ξψ ξ ξ ξ ξ ξ ξ ξ ′ ′′′ ′ ′ ′ ′ ′ ′′ ′ ′′′ ′′′ Π = - - + - + - + ′′ ′ ′′′ ′′ ′ + + + + + (2.19)
Given initial conditions, the governing equations (2.16, 17) can be solved numerically, which will be presented in section 4.

Explicit model for mode transformation

Equations (2.16, 17) were employed in [START_REF] Li | Intermittent Transformation between Radial Breathing and Flexural Vibration Modes in a Single-Walled Carbon Nanotube[END_REF] to study the intermittent transformation between RBM and flexural vibration mode in single-walled carbon nanotubes where the numerical instability was first observed without further investigation. Experimental observations show that, when a nearly uniform impulse is applied on a thin cylindrical shell, it may undergo a motion of intermittent vibration mode transformations, i.e. the extensional mode and flexural mode occur intermittently through a nonlinear coupling [START_REF] Zhu | Dynamic Response of Cylindrical Explosive Chambers to Internal Blast Loading Produced by a Concentrated Charge[END_REF][START_REF] Duffey | Strain Growth in Spherical Explosive Chambers Subjected to Internal Blast Loading[END_REF], Li et al., 2008). Since the circular cylindrical shell is in a closed form, the two non-dimensional variables ξ and ψ must be periodic with respect to angle θ and accordingly can be expanded in following Fourier series

0 1 2 ( ) ( ) cos ( ) cos n n n n a a n c n ξ τ τ θ τ θ ∞ ∞ = = = + + ∑ ∑ , ( 2.20) 1 2
( ) sin ( )sin

n n n n b n d n ψ τ θ τ θ ∞ ∞ = = = + ∑ ∑ . (2.21)
In above modal expansion of deformation variable ξ , only cosine terms are included since the mode transformations between pure extensional vibration mode (i.e. RBM) and the flexural modes represented by cosine terms are equivalent to the mode transformations between RBM and the flexural modes represented by sine terms. Accordingly only sine terms are included in the series expansion of ψ , which is due to the mathematical compatibility between the two expansions given in following Eqn. (2.22). A complete expansion with companion modes (sine terms) included can be found in (McIvor, 1966, Nayfeh and[START_REF] Nayfeh | Non-Linear Oscillation of Circular Cylindrical-Shells[END_REF]. In above expansions given by Eqns. Consequently, the four unknown generalized deformation coordinate functions n a and n b , n c

and n d can be reduced to two by the following relations ,

i i j j b ia i d c j j = - =   = =  L L (2.23) 1, 2,3, , 2,3,4,
Next, Lagrange equations will be used to produce governing equations of the unknown coefficient functions

( ) 0 a τ , ( ) n a τ and
( ) n c τ , for which kinetic energy T and potential energy U need to be given first. For a cylindrical shell of unit longitudinal length, the kinetic energy T and strain energy U , respectively, are given as ( )

2 2 2 2 2 2 2 1 0 0 1 1 1 2 2 r T ah r d E ah d t t π π ψ ρ θ ξ ξ ψ θ =   ∂ ∂       + = + -         ∂ ∂         ∫ ∫ & & (2.24)
and

( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 2 1 0 2 0 1 1 1 1 2 2 h h U z z a z d dz E ah a d π π θ θ σ ε θ ε α ε κ θ -   = + = + + -   ∫ ∫ ∫ . (2.25)
Substituting ε and κ given by Eqn.(2.14) and Eqn.(2.15) respectively into Eqn.(2.25), and then neglecting the fifth and higher order terms will yield

( ) ( )( ) ( ) { } 2 2 2 2 2 1 0 1 2 , 2 U E ah d π ψ ξ ψ ξ ξ ξψ α ξ ξ θ   ′ ′ ′ ′ ′′ = - + - - + + + ∆   ∫ (2.26)
where

( )( ) ( ) ( ) 2 2 2 2 2 2 2 0 1 2 2 4 d π α ψ ξ ξ ξ ξ ξψ ξ ψ ξ θ   ′ ′′ ′ ′ ′ ′ ∆ = - + + - - -     ∫ .
(2.27)

In the expression of strain energy in Eqn. (2.26), in which, since ∫ can be treated as a third-order small term. The term ∆ represents contributions to the degree of fourth order and is on at least one order smaller when compared to the second-and third-order terms for small deformation. As a result, the contributions from this term were discarded completely or partially in earlier publications [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF][START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF][START_REF] Lindberg | Stress Amplification in a Ring Caused by Dynamic Instability[END_REF]). However, it will be shown in next section that it is essential to include a complete fourth-order term in strain energy to guarantee the numerical stability of the solution.

The Lagrange equation for any generalized coordinate i q of a general mechanical system in absence of external loadings is 0, 1, 2,3,

i i i T T U i q q q τ   ∂ ∂ ∂ ∂ - + = =   ∂ ∂ ∂ ∂   L & . (2.28)
Substituting series of the two non-dimensional variables ξ and ψ given by Eqns. (2.20, 21) into energy integrals given by Eqns.(2.24, 26) respectively, then inserting the results of kinetic energy T and strain energy U into Lagrange Eqns.(2.28) while taking ( )

0 a τ , ( ) n a τ and
( ) n c τ as the generalized coordinate q i in sequence, will give the final governing differential equations of ( )

0 a τ , ( ) n a τ and ( ) n c τ , i.e.
( ) θ . (4) It is difficult to explicitly implement the integration in term ∆ when strain energy U is substituted into the Lagrange equation, therefore ∆ is kept in above governing equations representing the contributions from the fourth-order terms. As a result, it will recur to numerical method to carry out the integration in the term ∆ when numerical solutions of ( ) About the initial conditions for Eqns.(2.29-31), since we focus on mode transformations initiated from a nearly uniform RBM mode, the same initial conditions of ξ and ψ used in (McIvor, 1966) are introduced for convenience of comparison. Thus, the initial conditions are ( )

, 0 2 0 2 2 0 0 2 1 4 1 = ∂ ∆ ∂ + - - + ∑ ∞ a c n a a n n & & (2.29) ( ) ( ) ( ) [ ] ( ) [ ] ( ) [ ] { } , 0 1 1 2 2 2 5 2 1 1 1 ) 1 ( 1 2 1 2 2 2 0 2 2 2 2 2 2 4 1 = ∂ ∆ ∂ + + - - + + - + - + + + - + - + + + ∑ ∞ = - + n m n m m n m m n n n n n a n c c n m m c c n m m c n a n n a n c n n a n a α ( ) [ ] { ( ) [ ] }( ) . 0 1 2 ) ( 2 1 1 2 1 1 ) 2 ( 1 ) 1 ( 2 2 2 2 2 0 2 2 2 2 2 2 2 2 = ∞ = + - ∂ ∆ ∂ + + + - + + ∑ + + + - - + - + - + n m n m n m n n n c n n a m c n m n c n n c a n n n c n n n c m n m n α & & (2.
0 a τ ,
0 1 1 ,0 cos , n n v v n c ξ θ θ ∞ =   = +     ∑ & (2.32) ( ) ( ) ( ) 1 , 0 sin , , 0 , 0 0. v c ψ θ θ ξ θ ψ θ = - = = & (2.33)
In above initial conditions 0 v stands for a perfectly uniform radial velocity of the wall of the cylinder and

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 0 1 1 2 2 2 0 0, 0 , 0 0, 0 , 1 0 0, 0 , 2, 1 0 0, 0 , 2. 1 n n n n n n v v a a a a c c v a a n n c v n c c n n c  = = = =    = = ≥  +   = = ≥  +  & & & & (2.34)
The fourth order Runge-Kutta method (RK4) is employed to solve the differential Eqns. ( c ).

The numerical solutions of the explicit model will be given in section 3 with focus on the excitation of inextensional flexural modes and vibration stability while the numerical solutions of the implicit model will be given in section 4 with focus on stress growth. (2.32-34) will be defined for some specific cases. In this section explicit model in different cases will be introduced and discussed.

Numerical results of different explicit models

When the initial imperfections to uniform impulse are discarded completely, the cylindrical shell is assumed to be subjected to perfectly uniform radial impulse. Correspondingly, 0, 1 This solution indicates a vibration of simple in-and-out radial "breathing" mode (RBM) or pure extensional mode. Accordingly, cross-section of the cylindrical shell will remain circular at any instant. However, when imperfections are taken into account in the initial conditions, though their magnitudes are at least two orders smaller than that of the fundamental RBM vibration, numerical simulations in following sections will show that, as time increases, one or more inextensional flexural modes will be excited to an amplitude of the same order of RBM and the cylindrical shell will vibrate in coupled extensional and flexural modes. In following examples, in addition to the focus on the excitation of inextensional flexural modes, stability of the vibration, especially the effect of the fourth-order term ∆ on the stability of numerical solutions, will also be studied.

n v n = ≥ in

Goodier and McIvor's model

When the fourth-order term ∆ and the third-order term with coefficient 2 α are both omitted completely in the strain energy integral in Eqn.(2.26), and only inextensional flexural vibration is considered, i.e., 0, 43) and ( 44) in [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]. For this group of governing Eqns.(3.2,3), initial conditions of ξ and ψ in Eqns. (2.32, 33) should be modified as [Note: this is due to the necessity of mathematical compatibility. Detailed discussions can be referred to the reference [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]]

( ) ( ) ( ) ( ) 0 2 0 2 , 0 0, , 0 1 cos ,
, 0 0, , 0 sin , 0.01, 1, 2,3, .

n n n n n v n c v n n c n ξ θ ξ θ γ θ γ ψ θ ψ θ θ γ ∞ = ∞ =    = = +         = = ≤ =   ∑ ∑ & & L (3.4)
Accordingly, initial conditions for 0 a and n c can be determined as

( ) ( ) ( ) ( ) 0 0 0 0, 0 , 0 0, 0 , 1, 2,3, . n n n v a a c v c c n c γ  = =     = = =   & & L (3.5)
For a cylindrical shell with fixed aspect ratio a h , numerical results show that when the initial radial velocity 0 v reaches a sufficiently large value, one or more flexural vibration modes n c will be excited to amplitude of the same or even higher order of RBM 0 a . This phenomenon can be explained by the Mathieu equation and Mathieu stability diagram [START_REF] Mclachilan | Theory and Application of Mathieu Functions[END_REF][START_REF] Struble | Nonlinear Differential Equations[END_REF][START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF].

A numerical example is shown in Fig. 3 ). For the cylindrical shell under the initial condition ( 0 0.005 v c = ), it is shown in Fig. 3.1 that inextensional flexural mode c 6 is excited. Clearly, the cylindrical shell experienced a nonlinearly-coupled vibration of the extensional and flexural modes and the intermittent transformation of vibration modes and energy transfer happen [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF]. But, for the same cylindrical shell, when the initial radial impulse is stronger than some critical amplitude or equivalently when the initial velocity 0 v reaches a critical value, as discussed in [START_REF] Lindberg | Stress Amplification in a Ring Caused by Dynamic Instability[END_REF], numerical solution can easily begin divergent after a period of time, or the vibration can become instable soon. In [START_REF] Lindberg | Stress Amplification in a Ring Caused by Dynamic Instability[END_REF], this critical value of initial velocity 0 v was determined by the condition that stability parameter ( )

0 1 2 p v c α = = .
Then for the cylindrical shell shown in Fig. 3.1, the critical initial velocity can be determined as 0 0.00721 v c = . When the initial velocity reaches 0.0075c, the numerical results of RBM 0 a and inextensional flexural mode 6 c are shown in Fig. 3.2. It is evident that, with the increase of time, the periodic mode transformations between 0 a and 6 c start to fail and then the vibration becomes instable. Since this vibration instability still occurs at approximately the same moment (with slightly different vibration amplitude) when time step is refined from 0.1 to 0.01 (in non-dimensional time τ domain), it is reasonable to say that this vibration instability is not caused by numerical errors in the RK4 method, but due to the intrinsic physical instability in the formulation of the problem. However, if the fourth-order term ∆ in strain energy integral in Eqn.(2.26) is included together with the assumption that only one flexural vibration mode M c is possibly excited (actually from last example we know that 6 M = , and thus, only this specific flexural mode is included in the series expansions given in Eqn.(2.20, 21)), then the governing Eqns.(3.2,3) will be upgraded to ( )

2 2 2 0 0 0 1 1 2 0, 4 2 M M a a M c a c + - - + = && (3.6) 2 2 2 2 2 2 0 2 2 2 4 2 3 2 0 2 0. ( 1) ( 2) 1 1 3 4 4 1 8 3 M M M M M M M M M c c a c M M M M M c a c M α = - - + - + +     + - + +     +     && (3.7)
For the same problem shown in Fig. 3.2, numerical results based on Eqns.(3.6, 7) are shown in Fig. 3.3, which clearly shows that the cylindrical shell has undergone a stable motion for a much longer period with periodic vibration mode transformations. Therefore, it can be seen that the fourth-order term ∆ is crucial in the governing equations to guarantee a stable vibration solution.

McIvor's model

An explicit model with more generalized governing equations in [START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF] than those in [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF] was established to explore the growth of membrane and flexural stresses for a thin cylindrical shell under nearly uniform radial impulse. Similarly, the fourth-order term ∆ and the third-order term with coefficient 2 α were both discarded completely in the strain energy integral in Eqn. (2.26). The governing equations in the model [START_REF] Mcivor | Elastic Cylindrical Shell under Radial Impulse[END_REF]) is a special case of Eqns. (2.29-31). In the present study, in addition to the fourth-order term ∆, the term 0 n a a in Eqn.(2.30) was also dropped since extensional flexural terms n a are treated as at least one order smaller than inextensional flexural terms n c . With this group of governing equations, more general initial imperfections as described by Eqns. (2.32-34) can be investigated than the earlier model [START_REF] Goodier | The Elastic Cylindricall Shell under Nearly Uniform Radial Impulse[END_REF].

For this set of governing equations, Fig. 3.4 shows the numerical results of the same cylindrical shell under the same initial velocity 0 0.005 v c = as in Fig. 3.1. From Fig. 3.4, it can be seen that the cylindrical shell experiences a periodic transformations between the extensional and flexural modes in the early stage. However, as time τ passes 300, the shell begins to vibrate irregularly and becomes divergent very soon. Therefore, the vibration instability occurs again at a much shorter time compared with the stable period in Fig. 3.1 according to Goodier and McIvor's model.

It implies that the inclusion of extensional flexural mode imperfections has strong negative effect on the vibration stability.

Present explicit model

Though McIvor's model corresponds to a more general circumstance of initial impulse imperfections, numerical results show that when the fourth-order term is not included in the strain energy, the vibration can become instable soon after the impulse is imposed. However, it has been shown in Fig. 3.3 that stable vibration or steady intermittent mode transformations can be achieved when the fourth-order term is included in the governing Eqns. To further investigate the causes and nature of the numerical instability when the fourth-order term is omitted in the strain energy, numerical results of the two lowest vibration modes 2 c and 3 c and energy history for the formulations with and without the fourth-order term included in strain energy are presented in Figs.3.6 and 3.7, respectively. The dashed curves in Fig. 3.6 correspond to the case when the fourth-order term ∆ is not included in strain energy. The amplitudes of the two lowest modes 2 c and 3 c , which are supposed to be much smaller in amplitude than both the principle RBM 0 a and the major excited flexural mode 6 c , increase sharply to a level unbounded after certain period. This leads to an abrupt deviation of the RBM ( 0 a ) and the excited inextensional flexural mode ( 6 c ) from their periodic mode transformations.

When the fourth-order term ∆ is included in the strain energy integral in Eqn. (2.26), the predicted responses of 2 c and 3 c (solid curves in Fig. 3.6) are as expected to be periodic vibration with the amplitude at least one order smaller than that of 0 a and 6 c . This suppression of amplitude increase on low flexural modes can be explained by the energy histories with and without the fourth-order term included in the strain energy as shown in Fig. 3.7. Since there is no energy fluxed in or out of the whole system, the total energy of the system (kinetic energy T given by Eqn.(2.24) and strain energy U given by Eqn.(2.26)) should remain constant as shown by the solid curve in Fig. 3.7, which is predicted when the fourth-order term ∆ is included in the strain energy in Eqn. (2.26). However, the dashed curve in Fig. 3.7 shows that, when the fourth-order term ∆ is not included in the strain energy, the total energy E remains constant for a period of time before it departs dramatically from the initial value 0 E . By checking the strain energy expression in Eqn. (2.26), it is found that the fourth-order term is a key factor to guarantee the strain energy to be positively definite when the third-order term becomes negative and reaches the same order of the second-order terms after the inextensional flexural modes have been excited a few times during vibration.

The response at a typical point ( )

,

Numerical results of implicit model

For the original differential Eqns. (2.16, 17) of ξ and ψ , numerical solutions can be obtained by direct integration in the time ( ) τ domain with the fourth-order Runge-Kutta method and meanwhile using finite difference method in the space ( ) θ domain (use standard fourth-order central differences, [START_REF] Rappaz | Numerical Modelling in Materials Science and Engineering[END_REF]). In order to compare implicit model with the explicit model presented in last section, membrane stress m σ and flexural stress f σ at point ( )

, 0 a θ = are calculated with numerical solutions of ξ and ψ . With the definitions of normal force S and moment M given by Eqns. (2.11,12), membrane stress σ m is defined as the average stress on the side where normal force S is applied (see Fig. 2.2), and flexural stress σ f is defined as the maximum stress due to the moment M on the same side, i.e.

(

( ) 2 1 1 , 3 . 2 m f S E h M h E D σ ξ ψ α ξ ξ σ α ξ ξ  ′ ′′   = ≈ -+ +         ′′ = ⋅ ≈ +     (4.1) )
In order to compare predictions based on different models, exactly the same initial conditions are used equivalently for ξ and ψ , and for 0 a , n a and n c through Eqns. (2.32-34). All the following numerical results are obtained for the same cylindrical shell with aspect ratio 20.4 a h = but with varying initial RBM velocity ( ) 0 v in Eqns. (2.32-34). The initial velocity imperfections n v in Eqns. (2.32-34) are produced using the same way in previous section and fixed at two orders smaller than 0 v . Before we compare the numerical results from the two models, the second-order (without those underlined terms) and third-order (with those underlined terms) governing Eqns. at all during the period, especially in the last transformations. This difference can still be observed even when the time step is much smaller. It seems that the period of mode or stress transformation is stretched considerably longer in the third-order formulation of the problem. As energy is concerned, the total energy ( ) E values based on the two formulations are shown in Fig. 4.2. For an insulated system, the total energy should be constant. But the curves (i) and (ii) in Fig. 4.2 clearly show that energy deviates from its initial value E 0 around flexural stress peaks. It can be seen that the third-order formulation produces a much less energy deviation than the second-order one. This, together with the uniform pattern of stress transformations shown in Fig. 4.1(c) and (d), evidently indicates that the third-order formulation of implicit model provides satisfactorily higher precision than the second-order one. The energy curve in Fig. 4.2(ii) shows that the relative error of the numerical results from the third-order formulation is within 2%, hence thereafter all the numerical results based on the explicit model will use the third-order governing equations. predicted by the two models match well, and within each group of peaks, the frequency or period of stress vibrations is very close to each other, the phase-shifting phenomenon occurring in the second and subsequent groups of stress peaks can not be explained by the amplitude-frequency characteristics (AFCs) for free and forced vibrations of thin-walled structures [START_REF] Kubenko | Nonlinear Problems of the Vibration of Thin Shells (Review)[END_REF]. It is conjectured that the truncated series expansion of the deformation variable in explicit model limited the flexural modes to be coupled with the principle RBM, which leads to stronger coupling between the dominantly excited mode M c and the principle RBM 0 a . Hence, the energy loss from RBM 0 a will be fully transferred to M c as shown in Fig. 3.3. Therefore, the major excited flexural mode can get larger share of energy transferred from the RBM and reaches its peaks earlier. In early stage, the major excited flexural term is the dominant term and other flexural terms are so small that they can be neglected; the major excited flexural mode is almost identical to the deformation variable itself, which explains why the flexural stresses from two different models are almost identical in Fig. 4.4(c). However, with the increase of time, other minor flexural modes are gradually excited and their amplitudes are gradually increased. Their magnitudes, although still relatively low, become larger than their magnitudes in the early stage. Therefore, these minor flexural modes take more energy from the major excited flexural mode and cause the phase shift observed in Fig. 4.4(d). It is expected that this phase shift will become larger in further later group of stress peaks.

It should be noted that the inclusion of the fourth-order term in strain energy cannot guarantee the elimination of the numerical divergence in this vibration mode transformation problem. The reason is simply that, if the initial velocity is rather large, one or more flexural vibration modes (including lower flexural modes in general cases) may be excited, which may reach amplitudes higher than the RBM amplitude, resulting in non-positive definite strain energy. Theoretically, even higher-order terms may be required in the expression of strain energy in order to keep the strain energy positively-definite throughout the whole vibration or mode transformation process.

However, practically, the initial velocity is limited by the elastic limit of the shell material.

According to Eqn.(2.14), when RBM is considered in the early vibration phase of a thin-walled cylindrical shell, the circumferential strain is expressed by

a w = θ ε , (4.2)
where w is the radial displacement. Based on energy conservation, the relationship between the initial velocity and the maximum circumferential strain is , where y σ is the yield stress of the shell material. This value is normally less that 1% for engineering materials. Thus, we expect that the inclusion of fourth-order term in the present explicit model is sufficient for avoiding numerical divergence in most practical problems.

Conclusions

An implicit model and an explicit model in this analytical study are established with focus on the stability of vibration mode transformations between extensional and flexural modes in thin-walled cylindrical shells. Vibration mode transformation occurs when the initial radial velocity (or the impulse) is large enough. However, when the fourth-order term in strain energy is discarded, numerical solutions based on explicit model may become divergent, which is found to be associated with the non-positive definite strain energy when the negative third-order terms are accumulated to an amplitude close to the order of the second-order terms in strain energy. It is demonstrated that the inclusion of fourth-order terms in the strain energy function can practically eliminate the numerical instability in the vibration mode transformation in thin-walled cylindrical shells.

The same 2:1 internal resonance is revealed for the mode transformation based on implicit and explicit models. As flexural stress growth is concerned, the stress patterns predicted by both models match reasonably well. Two models predicted very close flexural stresses in the first group of peaks. However, in later group of stress peaks, phase shift gradually appears between the predicted flexural stresses based on implicit and explicit models. This can be explained by the

Figures:

Fig.

σ f /[E 1 (v 0 /c)] σ f /[E 1 (v 0 /c)] σ m /[E 1 (v 0 /c)] σ m /[E 1 (v 0 /c)]
σ f /[E 1 v 0 /c] σ f /[E 1 v 0 /c] σ f /[E 1 v 0 /c] τ σ f /[E 1 v 0 /c]

Figures:

Fig. ).

Fig. 2 .

 2 Fig.2.1, a point P on the undeformed midsurface of a thin cylindrical shell of radius a and thickness h moves to point P * . ( ) , a θ and ( ) , r φ are the polar coordinates of points P and P * ,

a

  where the dot above symbols indicates derivatives with respect to a new non-dimensional time variable t c a τ = ⋅

  d τ θ are both discarded since they correspond to rigid translation. Consider pure extensional and pure flexural deformation of a middle surface element independently, as shown in Fig.2.4, one can get the extensionality and inextensionality conditions,

  sought for governing Eqns.(2.29-31).

  n v n = L represent imperfections in this uniformity which are normally at least two orders smaller than 0 v . Correspondingly, the initial conditions for the governing Eqns.(2.29-31) of

In section 2 . 2

 22 an explicit model for mode transformation in cylindrical shell is established. When different imperfections are introduced in the initial conditions (correspondingly different flexural terms will need to be included in the model) or terms of different orders are included in the model, the explicit model given by governing Eqns.(2.29-31) combined with initial conditions in Eqns.

  are respectively the same as Eqn.(

  (3.6, 7), in which only the inextensional flexural imperfection is considered and only one inextensional flexural mode is assumed. In the present explicit model, numerical solutions for vibration and mode transformations will be sought based on the governing Eqns.(2.29-31) under the general initial conditions given in Eqn.(2.34). It should be noted that the fourth-order term ∆, which is included in Eqns.(2.29-31), is calculated according to Eqn.(2.27) with standard Gaussian integration method.The same cylindrical shell under the exactly same initial conditions (which means that the initial impulse imperfections are exactly the same) as those used in Fig.3.1 and Fig.3.4 is investigated with present explicit model and the results are shown in Fig.3.5. It can be seen that, compared with the vibration history based on McIvor's model as shown in Fig.3.4, stable vibration lasts more than twice the stable period predicted by McIvor's model and steady intermittent vibration transformations between RBM 0 a and the inextensional flexural mode 6 c can be observed.

  0 a θ = is investigated with the fourth-order term included in strain energy in Eqn.(2.26) and the results in frequency domain are shown in Fig.3.8. Two distinct frequency peaks can be seen in graph (c), which correspond to the RBM 0 a and the excited major flexural mode 6 c as plotted individually in insets in the graphs (a) and (b). It is shown that the frequency of the excited flexural mode is nearly half the RBM frequency, which can be explained by the 1:2 internal resonance between the major excited flexural mode and the principle RBM in thin-walled structures in the case of either free vibrations (Goodier and McIvor, 1964, Touze and Thomas, 2006) or forced vibrations (Thomas et al., 2005, Thomas et al., 2007, Amabili, 2008, Camier et al., 2009).

  (2.16,17) are used to predict the stresses and energy histories for the cylindrical shell under 0 0.005 v c = . It can be seen from Fig.4.1 that, the second-and the third-order governing equations predict quite different stress evolutions after the moment when the flexural stress reaches its maximum at time around 100 τ = [i.e. around the first group of peaks in Fig.4.1 (b) and (d)]. When compared to the uniform periodic transformation of membrane and flexural stresses during the total time span (600) shown in (c) and (d), the transformations of stresses shown in (a) and (b) occur at a higher frequency and are not uniform

Figure 4 .

 4 3 shows the frequency spectra of stresses at point ( ) third-order implicit model. Membrane stress, which is shown as inset in the spectrum (a) in Fig.4.3, should have the frequency of RBM since ψ ξ ′in Eqn.(4.1) is the dominant term in membrane stress and it corresponds to extensional vibration mode. Spectrum (b) in Fig.4.3 indicates that the flexural stress vibrates at half the frequency of membrane stress, which agrees with the frequency result given by the explicit model shown in Fig.3.8 since the flexural stress essentially comes from the flexural vibration. The flexural stress for the same cylindrical shell under 0 0.005 v c = is calculated to show the differences between the implicit model (third-order) and the present explicit model. Since the numerical solution based on the McIvor's model will become divergent after time τ slightly larger than 300, this model is not included in the comparison. The two graphs Fig.4.4(a) and (b) are the flexural stress responses from the implicit and explicit models, respectively. It can be seen that the flexural stresses produced by the two models follow very similar evolution patterns, especially that the first group of stress peaks (around time 100 τ = ) given by the two models shown in Fig.4.4(c) match very well. However, after the first group of peaks shown in Fig.4.4(c), the flexural stress based on the explicit model [dashed line shown in Fig.4.4(d)] reaches the subsequent peaks earlier than those based on the implicit model [solid line in the same graph]. It seems that, in the second group of peaks, the phase of the flexural stress predicted by the explicit model is shifted to the left in comparison with the phase of the flexural stress predicted by the implicit model. Since each pair of stress peak amplitudes among the first [Fig.4.4(c)] and second groups [Fig.4.4(d)]

  non-dimensional number c v 0 is limited by the yield strain of the material, i.e.

  limited number of terms included in the series expansions of explicit model. When only a limited number of terms are taken into account in the explicit model, compared to the inclusion of infinite number of terms in implicit model, the excited dominant flexural mode in the explicit model has a larger share of energy, and therefore, can be excited easily, especially in the later stage of vibration.

  Fig.2.1 A schematic representation of the deformation of a cylindrical shell with centroid fixed.

Fig. 2 . 2 AFig. 2 . 3

 2223 Fig.2.2 A typical cylindrical shell element: undeformed element (right) and deformed element (left).

Fig. 2 . 4 A

 24 Fig. 2.4 A schematic representation showing initial state (dash element) to current configuration (solid element) for a typical middle surface element. Arrows on both ends show the radial and tangential displacements, respectively.

Fig. 3 . 1 Fig. 3 . 2 Fig. 3 . 3 Fig. 3 . 4 Fig. 3 . 5 Fig. 3 . 6

 313233343536 Fig.3.1 Coupled motion of a cylindrical shell within Goodier and McIvor's model ( 20.4 a h = ,

Fig. 3 . 7

 37 Fig.3.7 Energy history of a cylindrical shell ( 20.4 a h = , 0 0.005 v c = ) with and without the

Fig. 3 . 8

 38 Fig.3.8 Response in frequency domain at a selected point of a cylindrical shell ( 20.4 a h = ,

Fig. 4 .

 4 Fig.4.1 (a) and (b), (c) and (d) are the membrane and flexural stresses of a cylindrical shell based respectively on the second-and third-order implicit model ( 20.4 a h = , 0 0.005 v c =).

Fig. 4 .Fig. 4 .

 44 Fig.4.2 (i) and (ii) are the total energy of a cylindrical shell based respectively on the second-and third-order implicit model ( 20.4 a h = , 0 0.005 v c =).

Fig. 4 .

 4 Fig.4.4 (a) and (b) are the responses of flexural stresses from the third-order implicit model (solid line) and present explicit model (dashed line), respectively. (c) and (d) are the superposed intercepts of them of the first and second group of stress peaks, respectively ( 20.4 a h = ,

  Fig.2.1 A schematic representation of the deformation of a cylindrical shell with centroid fixed.

Fig. 2 . 2 AFig. 2 . 3

 2223 Fig.2.2 A typical cylindrical shell element: undeformed element (right) and deformed element (left).

Fig. 2 . 4 A

 24 Fig. 2.4 A schematic representation showing initial state (dash element) to current configuration (solid element) for a typical middle surface element. Arrows on both ends show the radial and tangential displacements, respectively.

Fig. 3 . 1 Fig. 3 . 2 Fig. 3 . 3 Fig. 3 . 4 Fig. 3 . 5

 3132333435 Fig.3.1 Coupled motion of a cylindrical shell within Goodier and McIvor's model ( 20.4 a h = ,

Fig. 3 . 6

 36 Fig.3.6 Vibration histories of the two lowest flexural modes 2 c and 3 c of a cylindrical shell ( 20.4 a h = , 0 0.005 v c = ). Solid and dashed lines correspond to present explicit model and

Fig. 3 . 7

 37 Fig.3.7 Energy history of a cylindrical shell ( 20.4 a h = , 0 0.005 v c = ) with and without the

Fig. 3 . 8

 38 Fig.3.8 Response in frequency domain at a selected point of a cylindrical shell ( 20.4 a h = ,

Fig. 4 .Fig. 4 .Fig. 4 .Fig. 4 .

 4444 Fig.4.1 (a) and (b), (c) and (d) are the membrane and flexural stresses of a cylindrical shell based respectively on the second-and third-order implicit model ( 20.4 a h = , 0 0.005 v c =).

  between the two models because they are established following different approaches. But, regarding their accuracy, the explicit model may be one order higher in accuracy than the implicit model. Though the same basic strain expressions given by Eqns.(2.8, 14) are used in both models, truncated terms up to order four are kept in the strain energy in Eqn.(2.26), and therefore, in the final governing Eqns.(2.29-31) of the explicit model. However, truncated terms up to order only three are kept in the final governing Eqns (2.16, 17) of implicit model. But, it should be pointed out that the accuracy of the explicit model is sacrificed because only a limited number of terms can be included in the truncated series expansions in Eqns.(2.20, 21) when numerical solutions are sought. Furthermore, it is also adverse to the accuracy of explicit model where the extensional flexural terms ( n a ) in the final governing equations are discarded when they appear in summation together with the corresponding inextensional term ( n

	2.29-31)
	under initial conditions given in Eqn.(2.34).
	Now two sets of governing equations have been obtained for the cylindrical shell under radial

impulse: one is named the implicit model governed by Eqns.

(2.16,17)

and the other is called explicit model governed by Eqns.

(2.29-31)

. In general, the implicit model and explicit model for mode transformation in cylindrical shells are not strictly equivalent. In fact, it is difficult to give a precise equivalence

& & & &(2.17)

& &(2.30)

Acknowledgements

The author M.X. SHI acknowledges the financial support from Dorothy Hodgkin Postgraduate Awards (EPSRC-Shell Award).


Rafał P Piprek
email: rafal.piprek@uj.edu.pl

Anna Pecio

Jacek Z Kubiak

Jacek M Szymura

Differential effects of busulfan on gonadal development in five divergent anuran species

Keywords: Gonad, Testis, Ovary, Germ Cell, Busulfan

The effects of busulfan on the gonadal development were investigated in anuran amphibians.

The tadpoles treated with busulfan did not display sex reversal signs. The complete germ cell loss was observed in X. laevis. Germ cells are not necessary for the testis formation but are crucial during ovarian development.

Busulfan is an alkylating antineoplastic compound used as a common chemotherapeutic agent [START_REF] Bishop | Toxicological review of busulfan (Myleran)[END_REF]. In the intracellular environment, products of busulfan degradation bind the guanine bases of DNA and cause guanine-adenine intrastrand crosslinking [START_REF] Iwamoto | DNA intrastrand cross-link at the 5′-GA-3′ sequence formed by busulfan and its role in the cytotoxic effect[END_REF]. Such damage cannot be repaired and the affected cells undergo apoptosis, which may exert a teratogenic effect on the organism. Already in 1953, Bollag showed that the intraperitoneal administration of busulfan in rat caused germ cell depletion [START_REF] Bollag | The effect of myleran on rat gonads[END_REF]. Later, similar results were obtained in chicken and quail [START_REF] Reynaud | Germ cell population of chick embryos incubated 5 1/2 days after ultraviolet irradiation of young blastoderms[END_REF][START_REF] Reynaud | Effect of busulfan on the germ cell line of the quail embryo[END_REF][START_REF] Hallett | The effects of busulphan on gonadal differentiation and development in Japanese quail (Coturnix coturnix japonica)[END_REF]. This property is a serious drawback for busulfan use for human chemotherapy, where it causes gonadal failure resulting in a lack of sexual development and infertility [START_REF] López-Ibor | Gonadal failure following busulfan therapy in an adolescence girl[END_REF].

Gonads are unique organs because they are not necessary for somatic life but are crucial for reproduction and gene transfer from one generation to another. Embryonic gonads are composed of somatic cells (epithelial and mesenchymal cells) as well as germ cells that later give rise to the oocytes or spermatozoa. During embryogenesis, the germ cells originate as the primordial germ cells (PGS) in the regions distant from the embryonic gonads and then migrate towards genital ridges [START_REF] Nieuwkoop | Primordial Germ Cells in the Chordates: Embryogenesis and phylogenesis[END_REF][START_REF] Ogielska | The undifferentiated amphibian gonad[END_REF]. During migration, PGCs divide and for this reason they are very sensitive to the toxins such as busulfan [START_REF] Jackson | Effect of "Busulphan" ("Myleran") on the spermatogenic cell population on the rat testis[END_REF][START_REF] Merchant | Rat gonadal and ovarian organogenesis with and without germ cells. An ultrastructural study[END_REF]. In Xenopus embryo the PGCs are localized in the endoderm and afterwards actively migrate from the gut region through the dorsal mesentery to the genital ridges [START_REF] Nieuwkoop | Primordial Germ Cells in the Chordates: Embryogenesis and phylogenesis[END_REF].

The sexually undifferentiated gonads are composed of a cortex and medulla. The germ cells are incorporated into the cortex [START_REF] Witschi | Studies on sex differentiation and determination in amphibians: I. Development and sexual differentiation of the gonads of Rana sylvatica[END_REF]. During ovarian development, female germ cells (oogonia) remain in the cortex where they become enclosed by follicular cells.

During development of the testes the basal laminae between the cortex and medulla

A c c e p t e d M a n u s c r i p t 4 disintegrate and germ cells translocate into the medulla. The testis cords are formed within the central part of the gonad and are composed of Sertoli cells that enclose the germ cells (spermatogonia). The testis cords are rudiments of seminiferous tubules, within which spermatogenesis proceeds after the metamorphosis [START_REF] Ogielska | Spermatogenesis and male reproductive system in Amphibia-Anura[END_REF].

Several studies have examined gonadal development after germ cell depletion in various vertebrates [START_REF] Mclaren | Development of the mammalian gonad: The fate of the supporting cell lineage[END_REF][START_REF] Slanchev | Development without germ cells: The role of the germ line in zebrafish sex differentiation[END_REF][START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF]. In the zebrafish, ablation of germ cells caused transdifferentiation of the female gonad into the testis [START_REF] Slanchev | Development without germ cells: The role of the germ line in zebrafish sex differentiation[END_REF]. Similar female-to-male sex reversal has been observed in sterile mammals. The precursors of ovarian follicular cells transdifferentiate into clusters of Sertoli cells that form testis cord-like structures [START_REF] Merchant-Larios | Morphogenesis of the ovary from the sterile W/Wv mouse[END_REF][START_REF] Burgoyne | Perinatal oocyte loss in XO mice and its implications for the aetiology of gonadal dysgenesis in XO women[END_REF][START_REF] Guigon | Follicular cells acquire Sertoli cell characteristics after oocyte loss[END_REF].

On the other hand, the testicular development is not affected in the absence of germ cells and sterile seminiferous tubules are formed [START_REF] Mclaren | Development of the mammalian gonad: The fate of the supporting cell lineage[END_REF]. In birds such as quail, sex the germ cell ablation does not result in sex reversal [START_REF] Hallett | The effects of busulphan on gonadal differentiation and development in Japanese quail (Coturnix coturnix japonica)[END_REF]. Similarly, in the reptiles, such as the red-eared slider (Trachemys scripta) the germ cell depletion after busulfan treatment does not alter the sex of the gonads [START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF]. The effects of germ cell absence on amphibian gonads have been studied only in the early stages of genital ridge formation in X. laevis and Pelophylax esculentus [START_REF] Wylie | The formation of the gonadal ridge in Xenopus laevis. II. Scanning electron microscope study[END_REF][START_REF] Ogielska | Oogenesis and ovary development in the natural hybridogenetic water frog, Rana esculanta L. I. Tadpole stages until metamorphosis[END_REF]. These studies showed that the germ cells are not required for the formation of genital ridges.

Several studies showed that different vertebrates response differently to the loss of germ cells in the gonad [START_REF] Mclaren | Development of the mammalian gonad: The fate of the supporting cell lineage[END_REF][START_REF] Slanchev | Development without germ cells: The role of the germ line in zebrafish sex differentiation[END_REF][START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF]. Because our previous study showed that the gonadogenesis is different in divergent anuran amphibian species [START_REF] Piprek | Differentiation and development of gonads in the yellow-bellied toad, Bombina variegata L., 1758 (Amphibia: Anura: Bombinatoridae)[END_REF]. We assume that the effects of the germ cell depletion should vary within this group of vertebrates. Thus, we studied whether in different anuran species the gonadal sex is independent of the presence of germ cells and if the absence of germinal cells may lead , similar to zebrafish A c c e p t e d M a n u s c r i p t 5 and mouse, to sex reversal. To investigate this the 24h exposure of anuran tadpoles to water with 0.12 mM busulfan was carried out. We studied five anuran species representing distant phylogenetic lineages. The European fire-bellied toad (Bombina bombina: Bombinatoridae) and the African clawed frog (Xenopus laevis: Pipidae) are representatives of the most basal branches (Archaeobatrachia) [START_REF] Roelants | Global patterns of diversification in the history of modern amphibians[END_REF]. The more derived group, Neobatrachia, was represented by two sister lineages: Hyloidea (Bufo viridis: Bufonidae, Hyla arborea: Hylidae) and Ranoidea (Rana temporaria: Ranidae). Our choice was determined by the fact that the sex determination as well as the gonadal differentiation patterns vary considerably between these anuran species. In some species the males are heterogametic (Bombina sp., H. arborea, R. temporaria) whereas in others the females are heterogametic (X. laevis, B. viridis) [START_REF] Witschi | Studies on sex differentiation and determination in amphibians: III. Rudimentary hermaphroditism and Y chromosome Rana temporaria[END_REF][START_REF] Chang | Breeding of sex-reversed males of Xenopus laevis Daudin[END_REF][START_REF] Kawamura | Aspects of the reproductive biology of Japanese anurans[END_REF][START_REF] Engel | H-Y antigen as a tool for the determination of the heterogametic sex in Amphibia[END_REF]. Moreover, sexual differentiation of gonads can take place at various stages of development: in early larval period (Bombina sp., X. laevis, H. arborea), during the metamorphosis (B. viridis) or postmetamorphosis (R. temporaria) [START_REF] Piprek | Differentiation and development of gonads in the yellow-bellied toad, Bombina variegata L., 1758 (Amphibia: Anura: Bombinatoridae)[END_REF][START_REF] Eggert | Sex determination: the amphibian models[END_REF][START_REF] Ogielska | Pattern and rate of ovary differentiation with reference to somatic development in anuran amphibians[END_REF].

Materials and Methods

Animals

Larvae of X. laevis (n=139) were obtained in the laboratory whereas the eggs of B. bombina, H. arborea, B. viridis and R. temporaria were collected in the wild. The tadpoles were reared in 10-L aquaria. X. laevis larvae were fed with Sera Micron (Sera, Heinsberg, Germany) twice a day. All specimens used in the experiment were acquired according to Polish legal regulations concerning the protection of wild species (Dz. U. nr 33, poz. 289, 2005). We obtained permission from the Polish Ministry of Environment

Busulfan treatment

Busulfan (1,4-butanediol dimethanesulfonate (Sigma, Poznań, Poland)) was dissolved in DMSO (0.6 M stock solution) and 2 mL of stock solution was added to 10 L of dechlorinated water to the final concentration 0.12 mM (i.e. 30 mg/L). Tadpoles of X. laevis were staged according to Nieuwkoop and Faber [START_REF] Nieuwkoop | Normal Tables of Xenopus laevis (Daudin). 1st ed[END_REF] and the other species according to Gosner [START_REF] Gosner | A simplified table for staging anuran embryos and larvae with notes on identification[END_REF]. Larvae at stages proceeding the genital ridge formation, i.e. at the Nieuwkoop stage 45 for X. laevis or the Gosner stage 24 for the rest, were placed in 0.12 mM solution of busulfan for 24 h. Tadpoles kept in water with DMSO (0.2 ml/L) for 24 h were used as a negative control. Afterwards both busulfanized and control animals were reared in water without busulfan or DMSO at temperature of 19 o C and 12:12 L:D period. Premetamorphic larvae were staged and anesthetized with MS222 (Sigma, Poznań, Poland) solution once per week at Gosner stages 26, 30, 34, 37, 40, 44 and at Nieuwkoop stages 49, 51, 53, 55, 60, 64 for X. laevis. Postmetamorphic animals were anesthetized six months and one year after metamorphosis (Table 1). The gonads together with the kidneys and fat bodies were dissected and fixed in Bouin's solution.

Light microscopy

Fixed organs were dehydrated and embedded in paraplast (Sigma, Poznań, Poland). Then 6 m sections were stained with Debreuill trichrome [START_REF] Kiernan | Histological and Histochemical Methods: Theory and Practice[END_REF]. Images were taken with a Nikon Eclipse E600 light microscope and processed with Corel Photo-Paint A c c e p t e d M a n u s c r i p t 7 11. Numbers of cells were counted in 20 subsequent optical sections and compared using Student's t-test in Statistica 6 Pl software. The germ cells were recognized due to the large, pale nuclei and the Sertoli cells were defined as the cells located inward from the basal laminae of the testis cords or seminiferous tubules after germ cell ablation [START_REF] Russel | Effect of the microtubule disrupting agents, colchicines and vinblastine, on seminiferous tubule structure in the rat[END_REF].

Results

The influence of busulfan on anuran tadpoles

Busulfan treatment impacted anurans' survival as well as the shape and size of the body (Tables 1,2). The most noticeable effect was observed in the larval body length in B. bombina and B. viridis (Table 2). H. arborea larvae (37%) had impaired osmoregulation manifested by the storage of a large amount of fluid within the body. Some individuals (2%) showed malformations such as additional forelimbs. Because the busulfan had only a minor effect on the mortality and phenotype of larvae of X. laevis we have chosen this species to study the fate of germ cell depleted gonads.

3.2. Effect of busulfan on X. laevis gonads 3.2.1. Undifferentiated gonads X. laevis was the only species in which busulfan caused complete germ cell ablation, while the soma was almost unchanged (Table 3). Due to the disappearance of germ cells, the gonads of busulfanized individuals were smaller in comparison to the control (Tables 2,3). The formation of the genital ridges proceeded normally and began at the Nieuwkoop stage 49. The absence of the primordial germ cells during the formation of genital ridges indicated the apoptosis of PGCs during their extragonadal migration. At A c c e p t e d M a n u s c r i p t 8 the Nieuwkoop stage 51, the beginning of the medulla formation was visible due to the appearance of a somatic cell cluster in the gonadal hilus (the rudiment of medulla) (Fig. 1A,B). Despite of the absence of the germ cells the somatic cells of the gonad were more abundant than in control. At the Nieuwkoop stage 53, a well-separated medulla, which is the sign of the ovarian differentiation, was discernible in the center of gonad. The gonads devoid of germ cells had a larger amount of extracellular matrix in stromal space, i.e.

between the cortex and medulla. Melanophores and fibroblasts were visible within the relatively extensive stroma. The cortex and medulla of the ovary were lined with the folded basal laminae.

Ovary differentiation

At the Nieuwkoop stage 55, during sexual differentiation of the ovary, a secondary cavity appeared within the medulla due to dispersion of cells as in the control (Fig. 1C,D). In the absence of germ cells, the ovarian follicles or the germ cell nests in the cortex did not form (Fig. 1D). Somatic cells of the cortex were arranged in a thin layer covering the gonad and thus no follicular epithelium was observed. We did not identified any soma damages after busulfanization, i.e., exposure to busulfan, using the light microscopy (Fig. 1D). During the metamorphosis (the Nieuwkoop stage 64) the ovaries assumed the shape of a sac filled with an extensive cavity enclosed by two thin layers of somatic cells (cortex and medulla) separated by a thick sheath of extracellular matrix.

Testis differentiation

A c c e p t e d M a n u s c r i p t 9 At the Nieuwkoop stage 55, the sexual differentiation of the testis was recognizable owing to gathering of somatic cells into groups enclosed by the basal laminae in both the control and the busulfanized tadpoles (Fig. 1E,F). Thus the testis cords were formed regardless of the germ cell presence. The shrunken gonadal cortex transformed into the tunica albuginea enclosing the whole testis and a lumen appeared within the testis cords during metamorphosis as in the control. In the absence of germ cells, the seminiferous tubules were aligned with the Sertoli (epithelial) cells forming the pseudostratified epithelium. There were 22 ± 4.4 Sertoli cells per cross section of the testis cord in the busulfan-treated tadpoles whereas the control testis cords contained 5 ± 1.5 Sertoli cells per section (P<0.05; n=50). Thus the number of Sertoli cells increased over four times in comparison to the control, which was clearly visible in the histological sections (Fig. 1E,F). Typical connections between the testis tubules and the tubules of extragonadal system were observed in busulfanized and the control animals. Although the gonads attained a smaller size in comparison to the control, no signs of developmental retardation were noticed at the premetamorphic stages (Table 2).

Long term effects of busulfan

Six months after metamorphosis only a small remnants of testes were observed among extensive fat bodies in 7 among 12 busulfanized individuals (Fig. 2A). In the other 5 individuals no gonads were found at all. The number of the seminiferous tubules was drastically reduced within such residual testes and some abnormal vesicles were noticeable at their periphery. It can be supposed that the sterile seminiferous tubules transformed into such extensive vesicles in juveniles. One year after metamorphosis A c c e p t e d M a n u s c r i p t 10 gonads could not be detected during the macroscopic dissection of busulfanized frogs (n=10). The extremely large fat bodies (corpora adiposa) filled the space of abdominal cavity. The remnants of testes were found embedded in the fat bodies in 4 individuals among 10 studied and were composed of a few sterile testis tubules lined with monolayer epithelium (Fig. 2B). The fat bodies were attached along the entire lengths of the kidneys.

In 6 animals no signs of gonads could be noticed and no remnants of ovaries were discerned.

Effect of busulfan on the gonads in B. bombina

In busulfanized individuals of the European fire-bellied toad, gonads were distinctly smaller and retarded in development than the control (Table 2, Fig. 3A,B,C). A reduction in germ cell number was apparent in busulfanized individuals, however, these cells were not completely eliminated and 15.95% of germ cells survived until the Gosner stage 34 (Table 3). The genital ridges were formed at the Gosner stage 26 similar to the control. The cluster of somatic cells in the gonadal hilus appeared at the Gosner stage 30 (Fig. 3B), thus the rudimentary medulla was formed in spite of the germ cell absence. In the control gonads such a distinctive medulla was not noticeable at all (Fig. 3A). The visibly decreased number of somatic and germ cells resulted in alternation of the cortex and medulla differentiation and thus the connection of the medulla with the surface of the gonad, which was not observed in the control (Fig. 3A,B). The retardation of development was significant during both the pre-and postmetamorphic periods since the gonads found in six months old busulfan-treated toads were in the form of ridges composed of a thin sterile cortex and medulla and few gonial cells (Fig. 3C). In the gonads of busulfan treated larvae of Hyla arborea, the significant number of germ cells survived, i.e. up to 16.92% of germ cells until the Gosner stage 34 (Table 3, Fig. 3D,E,F). However, only 1.71% of germ cells survived until the metamorphosis, which may be a consequence of the impairment of the somatic part of the gonad. The size of the gonads in busulfanized tadpoles was similar to control (Table 2). The gonads in busulfanized tadpoles contained a larger amount of extracellular matrix distributed between germ and somatic cells (Fig. 3E). The separation of the cortex and medulla was not visible, which indicated a distortion of the cellular arrangement within the gonads after busulfanization (Fig. 3D,E). The presence of germ cells in both the cortical and medullar region, which is a sign of partial sex reversal, may resulted from the disruption of general structure of the gonad. Gonadal development after busulfanization was not visibly retarded in H. arborea before metamorphosis, which is probably related to the fact that the relatively high number of germ cells survived. After six months the gonads of busulfan-treated individuals were small and the medullary cells were embedded in abundant extracellular matrix (Fig. 3F). Some persisted gonial cells were present only in the peripheral region of the gonad.

Effect of busulfan on B. viridis gonads

The busulfan treatment of Bufo viridis resulted in the drastic decrease in the number of germ cells and the impairment of somatic cells (Table 3, Fig. 3H). Only 5.27% of germ cells persisted to the Gosner stage 34. The size of gonads was visibly reduced 2). Somatic cells in the gonads did not form cortico-medullary arrangement, which was present in the control (Fig. 3G,H). Six months after metamorphosis, only a streak of somatic cells persisted under the vena cava and there was no sign of the two gonadal layers in busulfanized individuals (Fig. 3I). Germ cells were encountered sporadically in such gonadal remnants.

Visible reduction of the germ cell number was the only sign of busulfan effect in the Bidder's organ. This organ is an ovary-like structure differentiated from the anterior part of the gonad in both males and females of all bufonids. The comparison of Bidder's organ structure before and after metamorphosis did not indicate the progressive degeneration after busulfan exposure, which is probably related to the high number of surviving germ cells (Table 3, Fig. 3J,K,L); about 25% of germ cells persisted in the Bidder's organ throughout the development (Table 3). Our observations showed a stronger cytotoxic effects of busulfan on the proper gonads than on the Bidder's organ.

Effect of busulfan on R. temporaria gonads

The genital ridges at Gosner stage 26 in busulfanized larvae of Rana temporaria were comparable to control. The number of surviving germ cells was highest among five tested species and was 16.92% at the Gosner stage 34 and significantly decreased to 7.24% around metamorphosis (Table 3, Fig. 3M,N,O). Before metamorphosis the corticomedullary arrangement of the busulfanized gonad was comparable to control. The amount of extracellular matrix was extremely large after busulfanization, nonetheless, the separation of the cortex and medulla was not evident (Fig. 3N). Persisting germ cells were located usually in the peripheral region of the gonad. Numerous small cavities were Busulfanized gonads just before metamorphosis resembled gonads of the Gosner stage 34 indicating a developmental retardation. Six months after metamorphosis, the gonads of busulfan-treated individuals were filled with the extensive space enclosed by a thick multilayer cortex containing germ cells (Fig. 3O). This indicates a strong retardation of the gonadal development after metamorphosis as well as the disappearance of the gonadal medulla.

Discussion

Busulfan is a potent anti-cancer agent, however, it has damaging side effect on normal cells. The most sensitive are proliferating primordial germ cells and therefore busulfan treatment leads to infertility. In amphibians busulfan treatment affects both germ and somatic cells of the gonad. In the majority of species investigated in this study (B. Structural distortion such as an excessive deposition of extracellular matrix visible in gonads of this species probably were the result of the germ cell loss during the development. X. laevis was the only tested species in which busulfan led to a total lack of germ cells without any degenerative effects on the seminiferous tubules and did not cause somatic damage in tadpoles resulting in the development of mature but sterile individuals. The specific resistance of the soma in X. laevis to busulfan may be related to its ploidy. This species is allotetraploid and evolved from interspecific hybridization [34], which can cause its unusual vigor. The persistence of gonadal structure despite the absence of germ cells as well as the high survival make X. laevis a good model species for studies of the role of germ cells on the gonad development. This model may facilitate the understanding of mechanisms responsible for germ cell aplasia in the human Del Castillo syndrome, also termed Sertoli cell-only syndrome (SCO). This syndrome is characterized by the total absence of germ cells in male patients (SCO type I) or by the presence of few germ cells in a minority of tubules (SCO type II). Patients with Del Castillo syndrome usually bear mutated Y chromosome, particularly deletions of the AZFa region containing gene USP9Y [START_REF] Castillo | Syndrome produced by absence of the germinal epithelium without impairment of the Sertoli or Leydig cells[END_REF][START_REF] Vogt | Human Y chromosome azoospermia factors (AZF) mapped to different subregions in Yq11[END_REF].

The most visible effect of busulfanization in anurans, besides germ cell loss, was the reduction of gonad size, a drastic developmental retardation and a tendency to gonad degeneration after metamorphosis. The small size of the gonads is the result of a lack of germ cells, similar to the Del Castillo syndrome in humans [START_REF] Castillo | Syndrome produced by absence of the germinal epithelium without impairment of the Sertoli or Leydig cells[END_REF]. We found that in all studied species the early genital ridges were normally formed. We observed the total lack of primordial germ cells during the formation of genital ridges only in X. laevis. This indicates that the primordial germ cell migration into the sites of gonadal formation did We found that in all studied species the cortex and medulla of the gonad began the development in spite of the absence of germ cells. Basal laminae appeared between these two gonadal parts. However, before metamorphosis the gonadal structure was disrupted, which was accompanied by the intensified deposition of extracellular matrix and basal lamina. We observed that the matrix was overdeveloped especially when a lot of somatic cells persisted. Thus the somatic cells are responsible for the formation of basal lamina and extracellular matrix components between the cortex and medulla and the germ cells are dispensable for the establishment of the cortico-medullary arrangement of the gonad.

The thick layer of stroma between the cortex and medulla did not inhibit the movement of germ cells since some surviving germ cells were often present in the gonadal medulla in R. temporaria and H. arborea. The presence of germ cells in the medulla suggests testicular differentiation whereas the appearance of a cavity within the medulla is a sign of ovarian development. We often observed the simultaneous occurrence of these two conditions in R. temporaria and H. arborea, which could be a sign of the partial sex reversal. Such an impairment of gonadal structure can be a result of a direct busulfan A c c e p t e d M a n u s c r i p t 16 effect on somatic cells rather than the germ cell loss. This is confirmed by the fact that the somatic part of the X. laevis gonad is almost unchanged in spite of the total germ cell lack.

Our data showed for the first time that the complete sex reversal is not observed after germ cell ablation in anuran amphibians. No signs of testicular differentiation were noticeable in developing anuran ovaries after germ cell ablation. Similarly, no sex reversal following the germ cell loss was observed in the slider Trachemys scripta [START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF].

Such resistance may result from better canalization of the sex determining pathway or from the lack of germ cell contribution to the sex determination in these species. Thus germ cells seem dispensable for sex determination in anurans, while they appear of key importance for the sexual differentiation of gonads in many vertebrates, e.g. in zebrafish (Danio rerio) and mammals. Namely, genetic depletion of germ cells in mouse and zebrafish leads to the development of testis structure in females following the trigger of the expression of male sex determination markers [START_REF] Slanchev | Development without germ cells: The role of the germ line in zebrafish sex differentiation[END_REF][START_REF] Siegfried | Germ line control of female sex determination in zebrafish[END_REF]. Some mutations leading to the germ cell loss followed by female-to-male sex reversal have been described also in the mouse [START_REF] Vainio | Female development in mammals is regulated by Wnt-4 signalling[END_REF][START_REF] Yao | Follistatin operates downstream of Wnt4 in mammalian ovary organogenesis[END_REF][START_REF] Piprek | Molecular mechanisms underlying female sex determination antagonism between female and male pathway[END_REF]. Likewise, busulfan treatment, in utero irradiation and long term-culture in vitro trigger the testicular development within the mammalian ovaries [START_REF] Guigon | Contribution of germ cells to the differentiation and maturation of the ovary: Insights from models of germ cell depletion[END_REF]. These observations indicated that germ cells in many mammals and zebrafish are critical for the maintenance of the female pathway in the ovary and repression of the male sex determination pathway. However, the structure of the ovary in X. laevis is probably more canalized and the male sex determination pathway is unable to take control over the female gonad and thus sex reversal does not occur.

The development of the testis in X. laevis was proceeding in spite of the germ cell A c c e p t e d M a n u s c r i p t 17 lack. Importantly, a lumen appeared among Sertoli cells forming seminiferous tubules in the testes during metamorphosis. This indicates that testicular differentiation and formation of the adult testis structure proceeds in spite of the lack of germ cells. It is also interesting that the number of Sertoli cells in the testes of busulfanized males increased in the absence of germ cells in X. laevis. Moreover, these testicular somatic cells were more numerous already since the beginning of gonadal development. A similar situation was observed in the testis of the slider T. scripta deprived of germ cells [START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF]. This implies that germ cells control the number of somatic cells in the gonad and that spermatogenic cells influence the somatic cells within the testis cords by inhibition of hyperproliferation of Sertoli cells.

During the ovarian differentiation in busulfan treated Xenopus tadpoles, as a result of germ cell absence, the somatic cells in the cortex formed only a thin layer. Similar condition was observed in busulfanized slider T. scripta [START_REF] Dinapoli | Germ cell depletion does not alter the morphogenesis of the fetal testis or ovary in the red-eared slider turtle (Trachemys scripta)[END_REF]. Neither epithelium of ovarian nests nor follicles were formed in busulfanized animals, suggesting that the development of these structures is induced and/or controlled by germ cells. This shows that germ cells are required in the ovary to form the follicular epithelium that together with oocytes constitute the ovarian follicles. In mammals, oocytes secrete many growth factors (such as GDF9) involved in the promotion of follicle cell proliferation and differentiation (folliculogenesis) [START_REF] Dong | Growth differentiation factor-9 is required during early ovarian folliculogenesis[END_REF]. Therefore germ cell depletion resulting in the complete lack of follicles in Xenopus indicates a similar mechanism of folliculogenesis in amphibians.

Surprisingly, degeneration of the gonads was proceeding after the metamorphosis in the individuals deprived of the germ cells. Only in H. arborea signs of gonad A c c e p t e d M a n u s c r i p t 18 degeneration was relatively mild, which suggested that the persisting germ cells prevent the progression of degeneration. The most significant degeneration of gonads after metamorphosis was observed in Xenopus . The females were probably deprived of the gonads altogether, however, the male gonads were small six months after metamorphosis and highly reduced in one year old frogs. The abdominal cavity of frogs was filled with extremely developed fat bodies (corpora adiposa) within which remnants of testicular tubules were found. Probably, the somatic cells of anuran gonads could transdifferentiate into fat cells (adipocytes) in the absence of germ cells, leading to the transition of sterile gonads into fat tissue after metamorphosis. We observed such a phenomenon exclusively in X. laevis. Thus it can be assumed that the persisting cells prevent the fat cell differentiation in other species. In anurans fat is stored in fat bodies that are formed from the anterior part of the genital ridges [START_REF] Ogielska | The undifferentiated amphibian gonad[END_REF]. This gonadal region loses its germinal function due to the germ cell loss and differentiates into fat bodies at early stages of gonadogenesis [START_REF] Ogielska | The undifferentiated amphibian gonad[END_REF][START_REF] Witschi | Studies on sex differentiation and determination in amphibians: I. Development and sexual differentiation of the gonads of Rana sylvatica[END_REF].It is possible that the molecular program of adipocyte differentiation is initiated in somatic cells in the absence of germ cells. Germ cells probably inhibit adipocyte-promoting factors such as insulin, IGF1, WNT10b, SHH, TGFβ, FGF, BMPs [START_REF] Rosen | Adipocyte differentiation from the inside out[END_REF]. Thus busulfanized X. laevis provides a good model for the study of the molecular machinery involved in fat tissue differentiation.

Conclusion

In summary, our research shows that in many anuran species the germ cells are unnecessary for: (i) the formation of genital ridges, (ii) the cortico-medullary differentiation of gonads, (iii) the sexual differentiation of gonads. However, busulfanization leads to various degree of impairment of gonad structures and the germ

  and approval from the I Local Commission for Ethics in Experiments on Animals.

A c c e p t e d M a n u s c r i p t 11 3. 4 .

 114 Effect of busulfan on H. arborea gonads

  gonadal centre instead of one large secondary cavity developing in the control. The somatic cells formed a thick peripheral layer covering the gonad.

  bombina, H. arborea, B. viridis, R. temporaria) the busulfan treatment resulted in a partial depletion of germ cells and somatic malformations as well as the increased mortality during metamorphosis. At early stages (i.e. Nieuwkoop stage 49 or Gosner stage 26) the animals were the most sensitive to busulfan, which was reflected in the highest mortality. The decreased survival was also observed at metamorphosis, which was also typical for control. The highest survival was characteristic for X. laevis while the highest mortality for B. viridis. In B. bombina, B. viridis and R. temporaria busulfan caused a significant impairment in the somatic part of the gonads and a retardation of their embryo development. In H. arborea the development of gonads was slightly altered.

  of the genital ridges. Several other studies also have shown that the genital ridges can develop in the absence of germ cells[START_REF] Wylie | The formation of the gonadal ridge in Xenopus laevis. II. Scanning electron microscope study[END_REF][START_REF] Ogielska | Oogenesis and ovary development in the natural hybridogenetic water frog, Rana esculanta L. I. Tadpole stages until metamorphosis[END_REF].Wylie and coworkers (1976) analyzed the earliest stage of X. laevis genital ridge development and concluded that only unorganized masses of somatic cells are formed when germ cells are depleted by UV-irradiation. However, this research did not extend to later stages of Xenopus development. The latter study[START_REF] Ogielska | Oogenesis and ovary development in the natural hybridogenetic water frog, Rana esculanta L. I. Tadpole stages until metamorphosis[END_REF] examined the interspecific hybrid Pelophylax esculentus in which germ cells often disappear during development and a few individuals exhibited an unaltered gonadal arrangement after germ cell apoptosis.

  be a key in the prevention of gonad degeneration after metamorphosis and/or their transition into fat bodies. The most resistant to busulfan treatment is X. laevis in which the somatic part of the gonad developed normally in the absence of germ cells.X. laevis may thus serve as a good model of Del Castillo syndrome for further studies of the molecular and cellular mechanisms participating in the function of the gonad deprived of germ cells.

Fig. 1 .

 1 Figure legends

Fig. 2 .

 2 Fig. 2. Sterile testes in Xenopus laevis after metamorphosis. A. Six months after

Fig. 3 .

 3 Fig. 3. Effects of busulfan on anuran gonads. A,B. The undifferentiated gonads in

Figure

  Figure

Acknowledgments

We are grateful to Dr. Malgorzata Kloc for valuable discussions and to Dr. M. Pabijan for English correction. This research was supported by a grant from the MNiSzW (N N303 542938). JZK was supported by a grant from ARC. RPP was supported by START stipend from FNP.