Nonlinear Shell Theory Research Articles

On the correspondence between two- and three-dimensional Eshelby tensors

We consider both three-dimensional (3D) and two-dimensional (2D) Eshelby tensors known also as energy–momentum tensors or chemical potential tensors, which are introduced within the nonlinear elasticity and the resultant nonlinear shell theory, respectively. We demonstrate that 2D Eshelby tensor is introduced earlier directly using 2D constitutive equations of nonlinear shells and can be derived also using the through-the-thickness procedure applied to a 3D shell-like body.

Analysis of the Problem of Stability of Thin Shells Compliant to Shear and Compression

The problem of stability of shells compliant to shear and compression is studied by the finite-element method. On the basis of relations of the geometrically nonlinear theory of thin shells compliant to shear and compression (six-mode version), we write the key equations for the determination of their initial postcritical state and formulate the corresponding variational problem. A numerical scheme of the finite-element method is constructed for the solution of the problems of stability of these shells. The order of the rate of convergence of the scheme proposed for the numerical solution of the problems of stability is investigated.

Methodology of studying non-planar films and membranes of complex structure

Among the thin-walled structural elements combining lightness and high strength, film and membrane elements are used most widely. Smart coatings with a complex structure are being actively developed nowadays. Proceeding from their functional duty, it is advisable to produce non-planar forms — shell films, membranes and coatings — which can have a complex structure specified by the designer or acquired in the process of manufacture and operation. Study of the mechanical properties of the shell films and membranes with a complex structure using standard uniaxial method of testing appeared ineffective. Complex structures with macro-heterogeneity should not be studied by indentation methods, capable of determining the material properties in the vicinity of the point in question. We developed an experimental-theoretical method for determination of the mechanical characteristics of non-planar films or membrane compositions of complex structure. At the stage of experiment, the stiffness of a non-planar sample (e.g., spherical, cylindrical or toroidal shape) fixed along the contour and loaded with by a unilateral surface pressure is estimated. Then, using the ratios derived from the nonlinear theory of shells, the integral mechanical characteristics of the shell sample material are determined: the reduced modulus of elasticity (elastic strain) or reduced conditional modulus of elasticity (plastic strain), deformation curves, etc. The relations for thin spherical membranes for the case of large displacements and deformations, as well as relations for thin cylindrical membranes of variable radius are considered. Results of the case study of rubber spherical membrane with holes and defect-free catenoidal shell are presented to illustrate the developed methodology.

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure.

We study the axisymmetric response of a complete spherical shell under homogeneous compressive pressure p to an additional point force. For a pressure p below the classical critical buckling pressure p_{c}, indentation by a point force does not lead to spontaneous buckling but an energy barrier has to be overcome. The states at the maximum of the energy barrier represent a subcritical branch of unstable stationary points, which are the transition states to a snap-through buckled state. Starting from nonlinear shallow shell theory, we obtain a closed analytical expression for the energy barrier height, which facilitates its effective numerical evaluation as a function of pressure by continuation techniques. We find a clear crossover between two regimes: For p/p_{c}≪1 the postbuckling barrier state is a mirror-inverted Pogorelov dimple, and for (1-p/p_{c})≪1 the barrier state is a shallow dimple with indentations smaller than shell thickness and exhibits extended oscillations, which are well described by linear response. We find systematic expansions of the nonlinear shallow shell equations about the Pogorelov mirror-inverted dimple for p/p_{c}≪1 and the linear response state for (1-p/p_{c})≪1, which enable us to derive asymptotic analytical results for the energy barrier landscape in both regimes. Upon approaching the buckling bifurcation at p_{c} from below, we find a softening of an ideal spherical shell. The stiffness for the linear response to point forces vanishes ∝(1-p/p_{c})^{1/2}; the buckling energy barrier vanishes ∝(1-p/p_{c})^{3/2}; and the shell indentation in the barrier state vanishes ∝(1-p/p_{c})^{1/2}. This makes shells sensitive to imperfections which can strongly reduce p_{c} in an avoided buckling bifurcation. We find the same softening scaling in the vicinity of the reduced critical buckling pressure also in the presence of imperfections. We can also show that the effect of axisymmetric imperfections on the buckling instability is identical to the effect of a point force that is preindenting the shell. In the Pogorelov limit, the energy barrier maximum diverges ∝(p/p_{c})^{-3} and the corresponding indentation diverges ∝(p/p_{c})^{-2}. Numerical prefactors for proportionalities both in the softening and the Pogorelov regime are calculated analytically. This also enables us to obtain results for the critical unbuckling pressure and the Maxwell pressure.

Analytical prediction of the impact response of graphene reinforced spinning cylindrical shells under axial and thermal loads

• An analytical study on impact responses of a cylindrical shell is presented. • Effects of the spinning motion, thermal load and external axial load on results are studied. • Reinforcement effect of graded GPLs is investigated. • A gratifying GPL distribution pattern for impact responses of cylindrical shells is carried out. This paper presents an analytical study that predicts the low-velocity impact response of a spinning functionally graded (FG) graphene reinforced cylindrical shell subjected to impact, external axial and thermal loads. The nanocomposite cylindrical shell is constructed based on a multiplayer model with graphene platelet (GPL) nanofillers whose weight fraction is constant in each concentric cylindrical layer but follows a layer-wise variation in the thickness direction, resulting in the position-dependent elastic moduli, mass density, Poisson's ratio and thermal expansion coefficient through the shell thickness. With effects of the thermal expansion deformation, external axial loads, centrifugal and Coriolis forces as well as the spin-induced initial hoop tension taken into account, the natural frequency of the cylindrical shell is derived on the base of differential equations of motion which are established according to the Donnell's nonlinear shell theory and the Hamilton's principle. The time-dependent contact force between a foreign impactor and the cylindrical shell is calculated by adopting a single spring-mass model. In addition, on the base of the other second-order differential equation, time-dependent displacements and strains are obtained by using the Duhamel integration. In numerical analyses, validation examples are carried out to verify the present solution, and then comprehensive parametric investigations are given to study effects of the GPL weight fraction, dispersion patterns, spinning speeds, temperature variations, geometrical sizes of the shell, the external axial load, radius of the impactor and the impact velocity on the contact force, contact duration and time histories of displacements and strains of the nanocomposite cylindrical shell.

Geometrically exact shell theory from a hierarchical perspective

A hierarchic approach for the derivation of an infinite series of nonlinear ell th-order shell theories from three-dimensional continuum mechanics based on a polynomial series expansion of the displacement field is recapitulated. Imposing the static constraints that second- and higher-order moments vanish, a ‘first-order’ shell theory is obtained without employing any kinematic constraints or geometric approximations. In particular, it is shown in full generality that, within the same theoretical framework, this static assumption, on the one hand, and a common Reissner–Mindlin-type kinematic assumption, on the other hand, lead to the same theory, for which the attribute geometrically exact is adopted from the literature. This coincidence can be interpreted as a theoretical justification for the heuristic Reissner–Mindlin assumption. Further, the unexpected but unavoidable appearance of transverse moment components (residual drill moments) is addressed and analysed. Feasible assumptions are formulated which allow to separate these drill components from the remaining balance equations without affecting the equilibrium of the standard static variables. This leads to a favourable structure of the component representation of balance equations in the sense that they formally coincide with the ones of linear shear-deformable shell theory. Finally, it is shown that this result affects the interpretation of applied boundary moments.

Mixed schemes of finite element method for non-standard boundary value problems of the nonlinear theory of thin elastic shells

Variational statements of equilibrium problems for a thin elastic shell within a geometrically nonlinear theory of mean bending for different types of principal boundary conditions, including non-classical, are given. Two classes of shell models are considered: (1) a geometrically nonlinear equilibrium model for an anisotropic shell of a material obeying generalized Hooke’s law; (2) a geometrically and physically nonlinear equilibria model for a shallow shell. Sufficient conditions for their generalized solvability in the corresponding Sobolev spaces are derived. In the case of the non-shallow shell the implicit function theorem is used. In the case of the shallow shell the variation problem is investigated using the generalized Weierstrass principle. Mixed finite element methods based on the use of the second derivatives of the deflection as auxiliary variables are constructed for approximate solutions of these problems. Sufficient conditions for solvability of the corresponding discrete problems are obtained. The convergence of approximate solutions is investigated. Accuracy estimates in the case of sufficiently smooth solutions of the original problems are given. Additional conditions are proposed for the problem on the shallow shell to ensure of implementation of the inequalities of the type of strong monotonicity and Lipschitz-continuity of the differential operator.

Influence of modal coupling and geometrical imperfections on the nonlinear buckling of cylindrical panels under static axial load

The aim of this work is to analyse the influence of the nonlinear modal coupling and initial geometrical imperfection on the post-buckling path (perfect case) or nonlinear equilibrium path (imperfect case) of a simply supported, axially loaded cylindrical panel. The cylindrical panel is described by the Donnell nonlinear shallow shell theory and the lateral displacement field is based on a perturbation procedure, generating a precise low-dimensional model that satisfies out-of-plane boundary conditions and considers the forthcoming nonlinear modal coupling due to quadratic and cubic terms in a nonlinear equilibrium equation. The discretized equations of motion are determined by applying the standard Galerkin method. Various numerical techniques are employed to obtain the cylindrical panel’s nonlinear static equilibrium path with its structural stability analysis. The results show the influence of geometry on the nonlinear response of the cylindrical panel, unveiling an intricate competition between different types of bifurcation diagrams (stable symmetric, unsymmetrical, and unstable symmetric). Completing the presented results, the initial geometrical imperfection could change the stability of the initial nonlinear equilibrium path (from unstable to stable) depending on the applied amplitude of the imperfection and its shape. It is possible to observe that the influence of the geometrical imperfection on the nonlinear equilibrium path of the imperfect cylindrical panel is determinant.

Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets

This paper performs nonlinear vibration analysis of metal foam circular cylindrical shells reinforced with graphene platelets. An improved Donnell nonlinear shell theory is employed to formulate the present model. The graphene platelet reinforced material properties are evaluated by the Halpin–Tsai equation. Different types of porosity and graphene platelet (GPL) distribution are taken into account. Governing equations are derived via Hamilton's principle and then they are transformed to ordinary differential equations using the Galerkin method. Afterwards, nonlinear frequencies of the system are solved by using the multiple scale method. Our findings demonstrate that GPL reinforced metal foam (GPLRMF) shells exhibit hardening-spring vibration characteristics. The nonlinear to linear frequency ratio of the shell closely relates to the porosity distributions and GPL patterns. The effect of geometrical size of graphene platelets on nonlinear vibration characteristics of GPLRMF cylindrical shells is also highlighted.

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis

We propose 2-D Cosserat type orthotropic constitutive equations for laminated shells for the purpose of initial failure estimation in a laminate layer. We use nonlinear 6-parameter shell theory with asymmetric membrane strain measures and Cosserat kinematics as the framework. This theory is specially dedicated to the analysis of irregular shells, inter alia, with orthogonal intersections, since it takes into account the drilling rotation degree of freedom. Therefore, the shell is endowed naturally with 6 degrees of freedom: 3 translations and 3 rotations. The proposed equations are formulated from the statement of the generalized Cosserat plane stress with additional transverse shear components and integrated over the shell's thickness using the equivalent single layer approach (ESL). The resulting formulae are implemented into the own Fortran code enabling nonlinear shell analysis. Some numerical results are presented to show the performance of the proposed approach.

Vibration characteristics of a rotating composite laminated cylindrical shell in subsonic air flow and hygrothermal environment

This paper focuses on vibration characteristics of a rotating composite laminated cylindrical shell subjected to both subsonic air flow and hygrothermal effects. Based on Love's nonlinear shell theory, and introducing hygrothermal strains into the constitutive relation of single layer material, the dynamic equations of the shell considering rotation, subsonic air flow and hygrothermal effects are obtained by Hamilton's principle. The frequency parameters of the equations are derived by means of Galerkin's method. Some numerical results are performed to conduct detailed parametric studies on vibration characteristics of the shell. In particular, combined effects of subsonic air flow and hygrothermal environment on natural frequencies of forward and backward travelling waves and critical rotating velocity of the shell are discussed, and the influence of initial hoop tension on those frequencies is also carried out. From the results it is shown that rotating angular velocity, subsonic air flow velocity and hygrothermal effects show the significant influence on vibration characteristics of the shell.

Bifurcation and Chaos of Piezoelectric Shell Reinforced with BNNTs Under Electro-Thermo-Mechanical Loadings

By employing the nonlinear von Karman shell theory and the theory of piezoelectricity including thermal effects, the constitutive relations of the BNNT-reinforced piezoelectric shell are built. Recurring to the ‘XY’ rectangle model, the material constants are reckoned. Then, the nonlinear governing equations of the structure are derived through the Reissner variational principle and solved by the fourth-order Runge–Kutta method. In numerical calculations, the effects of temperature, voltage, volume fraction, etc., on the bifurcation and chaos of piezoelectric shell reinforced with BNNTs are discussed in detail.

Nonlinear free vibration of graded graphene reinforced cylindrical shells: Effects of spinning motion and axial load

This paper presents an analytical study on linear and nonlinear free vibration characteristics and dynamic responses of spinning functionally graded (FG) graphene reinforced thin cylindrical shells with various boundary conditions and subjected to a static axial load. The weight fraction of graphene platelet (GPL) nanofillers in each concentric cylindrical layer is constant but follows a layer-wise variation through thickness direction, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the shell thickness. Based on the Donnell's nonlinear shell theory, the nonlinear partial differential equations of motion for the cylindrical shell are established by using the Hamilton's principle with the effects of centrifugal and Coriolis forces as well as the spin-induced initial hoop tension taken into account. The governing equations for nonlinear vibration of the nanocomposite cylindrical shell with different GPL dispersion patterns are derived from a set of nonlinear ordinary differential equations which are derived by employing the Galerkin approach. Dynamic responses of forward and backward travelling waves are investigated by analyzing the wave form and the frequency spectrum. Special attention is given to the effects of the weight fraction, dispersion patterns and the geometrical size of GPLs, the external axial load and spinning speeds of the cylindrical shell on the linear and nonlinear free vibrations of the nanocomposite cylindrical shell.

Nonlinear model of human descending thoracic aortic segments with residual stresses.

The nonlinear static deformation of human descending thoracic aortic segments is investigated. The aorta segments are modeled as straight axisymmetric circular cylindrical shells with three hyperelastic anisotropic layers and residual stresses by using an advanced nonlinear shell theory with higher-order thickness deformation not available in commercial finite element codes. The residual stresses are evaluated in the closed configuration in an original way making use of the multiplicative decomposition. The model was initially validated through comparison with published numerical and experimental data for artery and aorta segments. Then, two different cases of healthy thoracic descending aorta segments were numerically simulated. Material data and residual stresses used in the models came from published layer-specific experiments for human aortas. The material model adopted in the study is the mechanically based Gasser-Ogden-Holzapfel, which takes into account collagen fiber dispersion. Numerical results present a difference between systolic and diastolic inner radii close to the data available in literature from in vivo measurements for the corresponding age groups. Constant length of the aortic segment between systolic and diastolic pressures was obtained for the material model that takes the dispersion of the fiber orientations into account.

Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type

The paper is a continuation of [1] where the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter (the drilling degree of freedom) was presented. Here the formulation is extended to the elasto-plastic range. The material law is based on J2 Cosserat plasticity and employs the well-known Tamura-Tomota-Ozawa (TTO) [2] mixture model with additional formulae for Cosserat material parameters. Formulation is verified by solving a set of demanding analyses of plates, curved and multi-branched shells, including geometry, thickness and material distribution variation parameter analyses.

Simulation of Seismic Explosion Waves with Underground Pipe Interaction

Abstract The present paper provides numerical simulation of interaction of wave processes in the system “soil massif – underground pipeline” in explosion of charge on the surface of ground. Construction is considered in the framework of nonlinear theory of shells of Tymoshenko type. Soil is modelled by a solid porous multicomponent visco-plastic medium. The study establishes patterns of changes in stress-strain state of system depending on depth of pipeline laying and its diameter.

Progressive failure analysis of laminates in the framework of 6-field non-linear shell theory

The paper presents the model of progressive failure analysis of laminates incorporated into the 6-field non-linear shell theory with non-symmetrical strain measures of Cosserat type. Such a theory is specially recommended in the analysis of shells with intersections due to its specific kinematics including the so-called drilling rotation. As a consequence of asymmetry of strain measures, modified laminates failure criteria must be used in the prediction of the failure initiation. The model is implemented into the noncommercial numerical code of finite element method. Several examples are examined and the obtained solutions are compared with analytical, numerical and experimental results.

Smart damping of large amplitude vibrations of variable thickness laminated composite shells

This paper is concerned with the smart constrained layer damping (SCLD) treatment of laminated composite shells with variable thickness undergoing geometrically nonlinear vibrations. Three dimensional fractional derivative model (FDM) has been implemented for modelling the constrained viscoelastic layer of the SCLD treatment. The constraining layer of the SCLD treatment is made of vertically/obliquely reinforced 1–3 piezoelectric composites (PZCs) and acts as the distributed actuator. The strain-displacement relations are based on the simplified Novozhilov nonlinear shell theory to introduce the geometric nonlinearity in the large amplitude vibrations of the variable thickness shells. A three dimensional smart nonlinear finite element (FE) model has been developed for carrying out this analysis. Several numerical results are presented to check the accuracy of the present three-dimensional FDM for analyzing the passive and active control authority of the SCLD patch. Also the efficacy of the activated SCLD patch in controlling geometrically nonlinear vibration is computed for variable thickness shells and compared with shells of constant thickness.

Nonlinear Dynamics of Dacron Aortic Prostheses Conveying Pulsatile Flow.

This study addresses the dynamic response to pulsatile physiological blood flow and pressure of a woven Dacron graft currently used in thoracic aortic surgery. The model of the prosthesis assumes a cylindrical orthotropic shell described by means of nonlinear Novozhilov shell theory. The blood flow is modeled as Newtonian pulsatile flow, and unsteady viscous effects are included. Coupled fluid-structure Lagrange equations for open systems with wave propagation subject to pulsatile flow are applied. Physiological waveforms of blood pressure and velocity are approximated with the first eight harmonics of the corresponding Fourier series. Time responses of the prosthetic wall radial displacement are considered for two physiological conditions: at rest (60 bpm) and at high heart rate (180 bpm). While the response at 60 bpm reproduces the behavior of the pulsatile pressure, higher harmonics frequency contributions are observed at 180 bpm altering the shape of the time response. Frequency-responses show resonance peaks for heart rates between 130 bpm and 200 bpm due to higher harmonics of the pulsatile flow excitation. These resonant peaks correspond to unwanted high-frequency radial oscillations of the vessel wall that can compromise the long-term functioning of the prosthesis in case of significant physical activity. Thanks to this study, the dynamic response of Dacron prostheses to pulsatile flow can be understood as well as some possible complications in case of significant physical activity.