Nonlinearly Elastic Shells Research Articles

Efficient large-deformation shell elements for the simulation of electromechanical coupling effects in incompressible dielectric elastomers

So-called electro-active polymers are not only capable of undergoing very large deformations, but they also exhibit different electromechanical coupling effects due to their dielectric or electrostrictive characteristics. In the present paper the sub-class of dielectric elastomers is studied; in particular, we focus on thin dielectric elastomer shells, which undergo large deformations under an applied electric field. We develop a framework for the modeling and subsequent efficient simulation of thin structures made from incompressible dielectric elastomers. A thermodynamically consistent phenomenological continuum model based on the free energy density is adapted imposing the kinematic assumptions of Kirchhoff-Love shells in combination with including the thickness deformation and the electric field as independent unknown fields. To make the formulation accessible to finite element simulations, existing non-linear elastic shell mixed finite elements are extended to the present hyperelastic electromechanically coupled formulation for dielectric elastomer shells. Computational results of the shell theory are compared to three-dimensional results validating the proposed modeling and numerical simulation methodology, and showing high accuracy already for coarse finite element discretizations of the shell.

Formation of wrinkles on a coated substrate using manifold‐valued finite elements

AbstractThis article treats finite element simulations of controlled wrinkle formation experiments of a soft bulk material with a thin, stiff layer on top. The wrinkling process is triggered by a stress mismatch between the bulk material and the thin layer. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. We use Lagrange finite elements for the bulk material and geodesic finite elements for the shell. The resulting minimization problem is nonlinear and nonconvex. We prove existence of minimizers in the continuous and the discrete function space. Finally, we solve the resulting nonconvex minimization problem numerically using a Riemannian trust‐region algorithm and compare our simulations to real experiments.

A geometrically nonlinear Cosserat shell model for orientable and non-orientable surfaces: Discretization with geometric finite elements

We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by Bîrsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-folds or creases. The original model is generalized to shells that are not homeomorphic to a subset of R2. For this, we replace the originally planar parameter domain by an abstract two-dimensional manifold, and verify that the hyperelastic shell energy and three-dimensional reconstruction are invariant under changes of the local coordinate systems. This general approach allows to determine the elastic response for even non-orientable surfaces like the Möbius strip and the Klein bottle. We discretize the model with a geometric finite element method and, using that geometric finite elements are H1-conforming, prove that the discrete shell model has a solution. Numerical tests then show the general performance and versatility of the model and discretization method.

Intrinsic Marguerre–von Kármán equations

We first show how the classical Marguerre–von Kármán equations modeling the deformation of a nonlinearly elastic shallow shell can be recast as equations whose sole unknowns are the bending moments and stress resultants inside the middle surface of the shell. Thus, these equations allow to compute the stresses inside the shell without having to compute first the displacement field. We then show that the boundary value problem formed by these new equations is well posed by establishing an existence theorem.

Navier-Stokes-Fourier fluids interacting with elastic shells

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. % Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.

Existence results for the higher order linear Cosserat shell model

AbstractWe derive the linear elastic Cosserat shell model of order O(h5) in the shell thickness h. To this aim, we linearise the geometrically nonlinear elastic Cosserat shell model established previously. Using Korn‐type inequalities for shells, the coercivity of the energy functional and the Lax‐Milgram theorem, we prove the existence and uniqueness of weak solutions to the equilibrium equations.

A Linear Isotropic Cosserat Shell Model Including Terms up to $O(h^{5})$. Existence and Uniqueness

In this paper we derive the linear elastic Cosserat shell model incorporating in the variational problem effects up to order $O(h^{5})$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^{3})$ model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted.

Волны деформации в двух соосных, физически нелинейных оболочках с конструкционным демпфированием, взаимодействующих с окружающей средой и заполненных жидкостью

The paper investigates longitudinal strain waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid situated both between the shells and within the inner shell. The study takes into account the effects that the following factors have upon the magnitude and velocity of the wave: viscosity, fluid motion inertia, the medium surrounding the outer shell, and structural damping in the shell material. As it is impossible to evaluate the strain wave models proposed via qualitative analysis methods, using numerical methods is necessary. We numerically investigated the model presented using a specific difference scheme to represent its equations, one similar to the Crank --- Nicolson scheme for the thermal conductivity equation. We show that the velocities and amplitudes of strain waves in the shells do not change when disregarding the following factors: the fluid inside the shell, structural damping in the shell material, and the influence of the surrounding elastic medium. The waves evolve in the positive direction of the abscissa axis. At the same time, taking nonlinearity into account leads to increased wave velocity as compared to the linear case, when the wave propagation velocity equals the speed of sound, which means the waves become supersonic. We performed a numerical study of this case and obtained results matching the exact solution. If we take into account the effect that the fluid motion inertia within the inner shell has upon the wave velocity, then a decrease in the strain wave velocity is observed. Taking into account the presence of an elastic medium surrounding the outer shell leads to an increase in velocity. Considering the viscous properties of the fluid within the inner shell and the damping properties of the shell material results in decreased strain wave magnitudes. The models proposed can form the basis for designing modern non-destructive testing devices

An enhanced, rational model to study acoustic vaporization dynamics of a bubble encapsulated within a nonlinearly elastic shell

Acoustic droplet vaporization (ADV) is a new approach to generate vapor bubbles that have potentially broad medical applications. ADV-generated bubbles can be used as contrast agents in acoustic imaging, as drug carriers to deliver drugs to particular targets, and also in embolotherapy, thermal therapy, and histotripsy. However, despite much progress, ADV dynamics have still not been well understood and properly modeled. In this paper, we present a theoretical study of ultrasound-induced evaporation of a droplet encapsulated by a shell. The main emphasis of this theoretical study is on a proper description of the supercritical state occurring after bubble collapse. For this purpose, an isentropic equation of state for a van der Waals gas is used to describe the bubble behavior in the supercritical state. Sensitivity of the vaporization process is investigated for different acoustic and geometrical parameters and mechanical properties of the shell. Results show that the value of the minimum pressure required for direct vaporization (without any oscillatory behavior) depends on shell elasticity and initial size of the droplet, especially at high frequencies (greater than 2[MHz]). Moreover, it has been shown that applying an acoustic wave with proper phase such that thermal equilibrium of the bubble temperature with the surrounding liquid is attained, results in direct vaporization at lower acoustic pressure.

Axial tension of nonlinear elastic composite shells of zero Gaussian curvature with a rectangular hole

The formulation of physically nonlinear problems for composite shells of zero Gaussian curvature weakened by a rectangular hole under the action of axial loading is given. The initial equations are the equations of the theory of non-sloping shells, in which the Kirchhoff–Love hypotheses take place. Geometric relationships are written in vector form, and physical relationships are based on the deformation theory of plasticity for anisotropic materials. The system of resolving equations is obtained from the Lagrange variational principle. A technique has been developed for the numerical solution of two-dimensional physically nonlinear problems for orthotropic composite shells of this type, based on the use of the method of additional stresses and the method of finite elements. A variant of the finite element method is proposed, the peculiarity of which lies in the vector approximation of the sought values and the discrete execution of the geometric part of the Kirchhoff–Love hypotheses (at the nodes of finite elements). Using the developed technique, the nonlinear elastic state of an organoplastic conical shell with a rectangular hole, which at the ends is reinforced with frames and loaded with uniformly distributed tensile forces, has been investigated.

A Constrained Cosserat Shell Model up to Order $O(h^{5})$: Modelling, Existence of Minimizers, Relations to Classical Shell Models and Scaling Invariance of the Bending Tensor

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^5)$ and $O(h^3)$ case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya [Int. J. Solids and Struct., 2000] which will be further strengthened.

An Incompressible Polymer Fluid Interacting with a Koiter Shell

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

КОНЕЧНЫЕ ДЕФОРМАЦИИ ТОРОИДАЛЬНОЙ ОБОЛОЧКИ

The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.

The Isotropic Cosserat Shell Model Including Terms up to $O(h^{5})$. Part II: Existence of Minimizers

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solution for the theory including $O(h^{5})$ terms. The form of the energy allows us to show the coercivity for terms up to order $O(h^{5})$ and the convexity of the energy. Secondly, we consider only that part of the energy including $O(h^{3})$ terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including $O(h^{5})$ the conditions on the thickness $h$ are those considered in the modelling process and they are independent of the constitutive parameter, in the $O(h^{3})$ -case the coercivity is proven under some more restrictive conditions on the thickness $h$ .

The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells

On the basis of the smeared stiffener theory, the evolution of nonlinear axisymmetric longitudinal waves in a nonlinear elastic cylindrical shell reinforced by internal stringers is investigated. A non-classical nonlinear physical law is adopted for the shell material, which is characterized by a fractional degree of strain intensity. An analysis of the influence of an external elastic medium on a nonlinear wave process is carried out on the basis of the Winkler and shear-lag model. Using the asymptotic method of multiscale expansions, the Schamel-Ostrovsky quasi-hyperbolic equation for the component of longitudinal displacement was first derived. The Painlevé analysis of the equation showed the impossibility of exact solitary-wave solutions. The dispersion relation is analyzed, the maximum value of the phase velocity of infinitesimal disturbances is determined, above which stationary nonlinear waves can propagate. Based on the finite-difference approach and the Petviashvili method, a numerical simulation of the derived equation is carried out, profiles of stably propagating wave packets and solitary waves are constructed. A special dispersionless case, impossible for an unstiffened shell, has been identified, in which there are exact compactons, peakons and periodic solutions.

СВОБОДНЫЕ И ВЫНУЖДЕННЫЕ КОЛЕБАНИЯ ОБОЛОЧЕК РАЗЛИЧНОЙ ФОРМЫ С УЧЕТОМ ПЕРЕМЕННОСТИ ТОЛЩИНЫ ИЗ НЕЛИНЕЙНО-УПРУГОГО МАТЕРИАЛА

the article considers issues of shell vibrations, which are widely used in various industries and construction. Shells serve as elements of building structures with large spans.
 The chapter “Introduction” discussed issues related to the use of shells in all areas of industry, in aviation, rocket and space technology, railway transport, in the oil and gas industry and provides examples of shells for use in ceilings of circuses, stations, hangars; in industry: shells of rotation used as tanks, containers, columns, reactors, etc. In the section “Materials and research methods” free and forced vibrations of shells of variable thickness made out of nonlinear elastic material reviewed. In shell calculations the Kirchhoff – Law hypothesis was used. It was found that, during vibrations, the shells experience relative deformation of elongation and shear of the surface with coordinates (x, y), as well as bending and torsion strains. It is established that vibrations in the shells lead to a rotation of the main directions of elasticity and regarding to the adopted coordinate axis to angle θ, and the elastic constants of the material depend on the elastic constant Biy – the main directions of the nonlinear elastic shells.
 When solving the system of equations of motion of the shell relative to the displacements arising in it during vibration, based on theory R and various methods. The values of the dimensionless frequency parameter for a spherical shell taking into account changes in its curvature, variability and thickness of elastic properties are obtained.

Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave.

On material optimisation for nonlinearly elastic plates and shells

This paper investigates the optimal distribution of hard and soft material on elastic plates. In the class of isometric deformations stationary points of a Kirchhoff plate functional with incorporated material hardness function are investigated and a compliance cost functional is taken into account. Under symmetry assumptions on the material distribution and the load it is shown that cylindrical solutions are stationary points. Furthermore, it is demonstrated that the optimal design of cylindrically deforming, clamped rectangular plates is non trivial, i.e. with a material distribution which is not just depending on one axial direction on the plate. Analytical results are complemented with numerical optimization results using a suitable finite element discretization and a phase field description of the material phases. Finally, using numerical methods an outlook on the optimal design of non isometrically deforming plates and shells is given.

A higher order geometrically nonlinear Cosserat‐shell model with initial curvature effects

AbstractUsing a geometrically motivated 8‐parameter ansatz through the thickness, we reduce a three‐dimensional shell‐like geometrically nonlinear Cosserat material to a fully two‐dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed‐form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order O(h5) in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending‐curvature tensor, with coefficients depending on the initial curvature of the shell.

Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending–curvature tensor, with coefficients depending on the initial curvature of the shell.