Anisotropic Thin Shells Research Articles

Light scattering from the metallic nanowires with radially anisotropic cloaks

We discuss the scattering properties of the core–shell nanowires containing metallic core with nonlocal effects and radially anisotropic shell based on the nonlocal full-wave theory. The tunable extinction spectrum could be obtained by adjusting the anisotropy parameter and geometric size of the core–shell nanowires. The main peak resulted from transverse mode would be enhanced with the optimal anisotropy parameter. The subsidiary peaks due to the longitudinal mode become dense for the coated nanowires with thin anisotropic shell. Moreover, S-shaped and Z-shaped Fano resonances in the core–shell nanowires are presented under weak dissipation. These results will have potential applications in optical communications and ultra-sensitive detections.

Fundamentals and Processes of Fluid Pressure Forming Technology for Complex Thin-Walled Components

• New fluid pressure forming processes for tubes, sheets and shells were developed. • Methodology of adjusting stress state was invented to solve defects of forming thin-walled structures. • Outstanding applications in aerospace and automobile industry are presented. A new generation of fluid pressure forming technology has been developed for the three typical structures of tubes, sheets, and shells, and hard-to-deform material components that are urgently needed for aerospace, aircraft, automobile, and high-speed train industries. In this paper, an overall review is introduced on the state of the art in fundamentals and processes for lower-pressure hydroforming of tubular components, double-sided pressure hydroforming of sheet components, die-less hydroforming of ellipsoidal shells, and dual hardening hot medium forming of hard-to-deform materials. Particular attention is paid to deformation behavior, stress state adjustment, defect prevention, and typical applications. In addition, future development directions of fluid pressure forming technology are discussed, including hyper lower-loading forming for ultra-large non-uniform components, precision forming for intermetallic compound and high-entropy alloy components, intelligent process and equipment, and precise finite element simulation of inhomogeneous and strong anisotropic thin shells.

Numerical use of {[formula omitted]}-approximation method thermoelastic anisotropic thin shell theory equations represented in a special form

A linear uncoupled quasi-static theory of thermoelastic anisotropic thin shells of constant thickness is formulated. In the derivation of equations and boundary conditions, a variant of the {m,n}-approximation method, which is based on the variational principle of the thermoelasticity theory, is used. According to this method, the unknown functions are represented in the form of a series of Legendre polynomials in the transverse coordinate that agree with the force boundary conditions on the facial surfaces. From the system of equations of the constructed theory for generalised displacements, strains, and stresses, a system of equations for generalised displacements is obtained; this system is solved for the second derivatives of generalised displacements with respect to one of the Gaussian parameters of the middle surface. For a system of partial differential equations represented in such a form, the known techniques for the reduction of systems of two-dimensional equations to normal systems of ordinary differential equations, which can be solved by standard numerical methods, can be applied. The system of equations for generalised displacements is reduced to a normal system of ordinary differential equations in the case of a plane deformation of a cylindrical panel. Through the use of these equations and the S. K. Godunov method of orthogonal successive substitutions, the bending problems of a planar beam subjected to a mechanical load or a thermal load and a circular cylindrical panel exposed to a thermal load are numerically solved. It is demonstrated that the proposed equations are particularly suitable for the analysis of boundary effects in planar beams and circular cylindrical panels.

Theory of the multilayer thin anisotropic shells, based on the asymptotic analysis of the general equations for the elasticity theory

A new theory of thin multilayer anisotropic elastic shells is proposed, based on the application of the asymptotic analysis over small geometric parameter to the general 3-dimensional equations of the elasticity theory for curvilinear coordinates. In deriving the equations of the shell theory no assumptions are made concerning to the distribution of displacements, deformations, or displacements trough thickness. Recurrent consequences of local problems of the elasticity theory for shells are formulated and an analytical solutions are obtained. The averaged equations of the asymptotic theory of shells that are of the same type as the classical equations of the Kirchhoff-Love theory of shells are derived. It is shown that the asymptotic theory makes it possible to obtain in explicit form the distributions of all 6 components of the stress tensor over the thickness of the shell. An example of the calculation of a cylindrical shell under axisymmetric bending is considered. The case of monoclinic shell layers, which have no more than 13 independent elastic constants, is considered. An algorithm is proposed for obtaining explicit analytical equations for calculating the distribution of components of the total stress tensor over the cylindrical shell. The effect of the geometric d imensions of the shell on the character of the distribution of displacements and stresses in the shell is analyzed.

Equations of Thin Anisotropic Elastic Shells of Revolution in the {m, n}-Approximation Method

We construct a system of differential equations aimed at the description of elastic deformations of thin anisotropic shells of revolution and solved with respect to the first-order partial derivatives with respect to the meridional coordinate. These equations are obtained by the {m,n}-approximation method. The approximations of unknown functions are in good agreement with force boundary conditions imposed on the front faces.

A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy

This work presents a simple finite element implementation of a geometrically exact and fully nonlinear Kirchhoff---Love shell model. Thus, the kinematics are based on a deformation gradient written in terms of the first- and second-order derivatives of the displacements. The resulting finite element formulation provides $$C^1$$C1-continuity using a penalty approach, which penalizes the kinking at the edges of neighboring elements. This approach enables the application of well-known $$C^0$$C0-continuous interpolations for the displacements, which leads to a simple finite element formulation, where the only unknowns are the nodal displacements. On the basis of polyconvex strain energy functions, the numerical framework for the simulation of isotropic and anisotropic thin shells is presented. A consistent plane stress condition is incorporated at the constitutive level of the model. A triangular finite element, with a quadratic interpolation for the displacements and a one-point integration for the enforcement of the $$C^1$$C1-continuity at the element interfaces leads to a robust shell element. Due to the simple nature of the element, even complex geometries can be meshed easily, which include folded and branched shells. The reliability and flexibility of the element formulation is shown in a couple of numerical examples, including also time dependent boundary value problems. A plane reference configuration is assumed for the shell mid-surface, but initially curved shells can be accomplished if one regards the initial configuration as a stress-free deformed state from the plane position, as done in previous works.

Stress Analysis for the Orthotropic Pressurized Structure of Cylindrical Shell with Nozzle

The structure of cylindrical shell with nozzle has been widely used in pressure equipment. The analysis and design for the structure of cylindrical shell with nozzle are generally based on the isotropy of metal at present, which may cause certain error and risk. Because of the approaches of rolling, heat treatment, and so on, the materials’ properties are different along each direction in practical engineering. In this paper, based on anisotropic thin shell theory, the analysis for orthotropic pressurized structure of thin-walled cylindrical shell with nozzle was conducted by using finite element software Ansys. The influences of thickness ratio and elastic modulus ratio were comprehensively considered, and also compared with the stress results under isotropic structure. The analysis results show that the values of stress and stress concentration factor for the structure of cylindrical shell with nozzle decrease with elastic modulus ratio, but increase with thickness ratio. Compared with the influence of thickness ratio, the elastic modulus ratio has significant effect on stress level. The rational use of the material anisotropic property can optimize the structure of thin-walled cylindrical shell with nozzle. The results in present paper provided a reference for the design of orthotropic pressurized structure of cylindrical shell with nozzle.

Drawing simulation by static implicit analysis with the artificial damping method

Wrinkling during draw is typically a local instability problem. When the structural instability is localized, there will be a local transfer of strain energy from one part of the structure to neighboring parts, and global solution methods, which is typically represented by the arc length method, may not work. So, this type of problems has to be solved either dynamically or with the artificial damping. On the other hand, the essential nature of the buckling behavior can be regarded as a static problem, even though it may be possible to raise some side issues due to the inertia effect. In this study, we traced the local buckling behavior of anisotropic elasto-plastic thin shells in Numisheet2014 BM4 using the artificial damping method.

Design by analysis for orthotropic pressurized structure with small end of conical shell and cylindrical shell based on Hill48 yield criterion

Through the discontinuity analysis of orthotropic pressurized structure with small end of conical shell and cylindrical shell based on anisotropic thin shell theory, the solution of shear, bending moment and stress at the juncture of conical shell and cylindrical shell have been derived. By using of Hill48 yield criterion and stress classification principles in the design of cone shell structure, stress solutions were obtained in this paper. The results show that the thickness is determined by local membrane stress for the small end of conical shells. The stress level is of little influence and can be ignored for an example in present paper, while yield criterion is great important. Higher yield stress of circumferential direction can improve the load capacity, the thickness of conical shell for orthotropy I mode can decrease about 10% compared with isotropic conical shell.

Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

A mixed variational principle is derived by Hamilton’s method from the principle of minimum potential energy for thin anisotropic shells of revolution and is then used to derive a normal system of equations with complex coefficients. Discrete orthogonalization is used to solve this homogeneous system and the nonlinear system of equations that describes the precritical state of shells. A shell generated by revolving a circular arc around the axis parallel to its chord is analyzed for stability. The solution is compared with the approximate solution obtained assuming that the precritical state is membrane. It is established that the approximate problem formulation gives incorrect results for shells of negative Gaussian curvature

Canonical equations in the geometrically nonlinear theory of thin anisotropic shells

The functional in the principle of minimum potential energy of layered anisotropic shells with a nonlinear relationship between strains and displacements is transformed into a canonical integral that coincides with the functional in the Reissner principle. Partial forms of the functional are derived for problem formulations where the dimension can be reduced with respect to one of the coordinates. The canonical system of equations is linearized and then normalized. The boundary-value problem is solved by the numerical discrete-orthogonalization method. An anisotropic spherical shell under external compression is analyzed for stability as an example

Vibration Analysis for Elastic Vascular Stent

A mathematical modeling for breathing vibration problem of vascular stent is presented. The vascular stent is considered in structure as an elastic cylindrical lattice shell, in which the cell element is treated as a spatial frame structure. Based on the force equilibrium and the deformation geometry analysis for the cell element, calculative formulae for effective elastic parameters of the lattice shell are obtained by using homogenization method. Hence, the vascular stent is modeled as an orthogonally anisotropic thin shell. The equation of vibration for vascular stent is derived from Flügge shell theory. The computation of natural frequency of vibration is performed. The analytic results from the presented formulae are close to that from finite element method by the means of ANSYS, which validates the applicability of the presented modeling of vascular stent structures. The vibration analysis could be a useful reference for practical engineering designs of vascular stent.

Construction of anisotropic polyconvex energies and applications to thin shells

In computer simulations where constitutive equations are considered anisotropic polyconvex energies can preferably be used because the existence of minimizers is then automatically guaranteed. In this work we investigate the capability to simulate anisotropy effects of anisotropic thin shells using polyconvex anisotropic energies. The construction of the considered polyconvex transversely isotropic energy is based on specific structural tensors. The iterative enforcement of the zero normal stress condition at the integration points allows the consideration of arbitrary three-dimensional constitutive equations. As a representative example we compare results for isotropic and anisotropic plates.

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell.

Polyconvex energy densities with applications to anisotropic thin shells

AbstractThis work shows the capability to simulate anisotropic thin shells using polyconvex energy densities by analyzing numerical examples. The variational framework is based on the enhanced assumed strain formulation. The iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three–dimensional constitutive equations. We consider an additive structure of the energy decoupled in an isotropic part for a matrix material and the superposition of transversely isotropic parts for embedded fiber families and focus on polyconvex strain energy functions. In two representative examples we document anisotropy effects. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of thin shells using anisotropic polyconvex energy densities

In this contribution the numerical framework for the simulation of anisotropic thin shells is presented on the basis of polyconvex strain energy functions. The nonlinear shell theory is based on the Reissner–Mindlin kinematic along with inextensible director vectors. For the variational framework we consider a three field variational functional taking into account independent displacements, enhanced strains and stress resultants, where the latter field is eliminated by the evaluation of some orthogonality conditions. The iterative enforcement of the zero normal stress condition at the integration points allows consideration of arbitrary three-dimensional constitutive equations. Due to the fact that we are interested in fiber-reinforced materials we consider an additive structure of the energy decoupled in an isotropic part for the matrix and a superposition of transversely isotropic parts for the fiber families. For the representation of the anisotropy we use the concept of structural tensors and formulate the strain energy function in terms of principal and mixed invariants of the right Cauchy–Green tensor and the structural tensor. In order to guarantee the existence of minimizers we focus on polyconvex strain energy functions. The anisotropy effect is documented in several representative examples.

Axisymmetric actuation of composite cylindrical thin shells with piezoelectric rings

Axisymmetric static and frequency analyses of anisotropic cylindrical thin shells with one and two perfectly bonded ring piezoactuators are performed. The shell is assumed to be linear elastic and made of laminated composite materials. The electroelastic constitutive relations for the piezoelectric materials are also assumed to be linear. It is shown that, if there exists a special relationship involving the membrane and the membrane-bending coupling stiffness matrices, the analysis is greatly simplified. In such a situation, simple closed form solutions of the equilibrium equations are obtained for the case of an infinite shell with one or two actuators. Kirchhoff's assumptions are used for the analysis and the dynamic formulation is derived from a variational principle which includes the total structural potential energy and the electrical potential energy of the piezoelectric material, involving both mechanical and electrical variables. The finite element method is then applied to obtain the stiffness and mass matrices. The computer code developed to implement the formulation allows the static and dynamic analyses of arbitrary cylindrical shells with piezoelectric actuators. Good agreement is reached between the analytical solution found and the numerical procedure implemented. Results indicate that the maximum normalized displacement and its location vary according to the actuator length. Furthermore, a frequency analysis is carried out in a broad range of frequencies to investigate the effect of mass properties on the response of a simply supported cylindrical shell.

A theory on anisotropic piezothermoelastic shell laminates with sensor/actuator applications

Piezoelectric materials have become very popular in dynamic sensing, actuation and control of active structural and mechanical systems in recent years. This paper sets out to provide a comprehensive thermo-electromechanical theory of anisotropic piezoelectric thin shell laminates subjected to mechanical, electric and temperature excitations. Piezothermoelastic constitutive equations of anisotropic piezoelectric materials are defined, and governing thermo-electromechanical equations and boundary conditions are derived using Hamilton’s principle. In general, the resultant forces/moments and electric displacements have three components, contributed by the elastic, electric and thermal fields, respectively; they interact in the system thermo-electromechanical equations and boundary conditions. Applications of the theory to dynamic sensing and control are also discussed. Due to the generalities of the material and geometry, the derived theory can be widely used in many piezoelectric materials, e.g., piezoceramics, piezo-polymers, etc., and also geometries, e.g., shells, plates, rings, beams, etc. Application examples are given in case studies.

Effect of mechanical anisotropy of flexible media on reproduced output

The realization of excellent head-to-tape interface is important to achieve high recording density in VCRs. Especially, the anisotropy of mechanical characteristics of the magnetic tape influences the head-to-tape contact. In the present study, we have measured the anisotropy of Young's modulus of magnetic tapes by measuring the ultrasonic propagation velocity. The anisotropy of the particulate medium differed from that of evaporated medium. We have made a computational calculation, by the Finite Element Method, of the deformation of the tapes having anisotropy of mechanical characteristics and sliding over a head fixed on a rotating drum. The flexible tape is modelled as a thin anisotropic shell and the air film is treated by the modified Reynolds equation. The lateral tape deformation affects the reproduced output.

Small‐Angle light scattering from an anisotropic sphere: Anisotropy and size effects

AbstractThe recently developed theory of light scattering from an anisotropic sphere (taken as a model for a spherulite) summarizes all information obtainable from such scattering system when the Rayleigh‐Gans‐Debye approximation is appropriate. The commonly used procedure of size determination from the maximum of the Hv intensity is tested for a number of models differing in radius size, inherent anisotropy δn and mean refractive index ¯n. It has been confirmed, that the parameter (¯n ‐ 1)/δn plays the decisive role in the angular dependence of the Hv intensity and the influence of that parameter on the size determination and resulting accuracy is specified. The results differ for positive and negative spherulites and the stabilizing role of increasing spherulite size in accuracy of procedure is demonstrated. It has been shown that the RGD approximation is appropriate especially for the case of homogeneous spherulite texture (thus implying, that the refractive index of the surrounding is close to the average refractive index of a single spherulite). The results extrapolated outside of the RGD approximation display invariable Umax position for isotropic contribution with increasing sphere size in contrast to the oscillatory character of Umax predicted by the LM solution. It is proposed to construct the anisotropic solution under an approximation recently suggested for thin anisotropic shell. © 1994 John Wiley & Sons, Inc.