Thin Elastic Bodies Research Articles

Numerical investigation of the 2d–1d structure interaction model

In this article, we explore the possibility of modeling the interaction of a 2d elastic body with a thin 2d elastic body of possibly higher thickness using a 1d model for the thin body. We use the asymptotic analysis with respect to the small thickness of the 2d interaction model and formulate five different limit models depending on the order of stiffness of the thin body with respect to the thickness. Then we formulate a 2d–1d model which has the same asymptotics as the 2d–thin 2d model with respect to thickness. Finally, we numerically test the approximation of the 2d–thin 2d model by the 2d–1d model on two problems, one with an analytical solution and one more realistic problem.

Non-classical aspects of Kirchhoff type shells

Kirchhoff type shells are continuum models used to study the mechanics of thin elastic bodies; these are largely based on the theory of surfaces. Here, we report a reformulation of Kirchhoff shells using the theory of moving frames. This reformulation permits us to treat the deformation and the geometry of the shell as two separate entities. The structure equations which represent the familiar torsion and curvature free conditions (of the ambient space) are used to combine deformation and geometry in a compatible way. From such a perspective, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of dislocations and disclinations). Using the proposed framework, we solve a boundary value problem and thus demonstrate, to an extent, the importance of moving frames.

The Whitham equation for hydroelastic waves

A weakly nonlinear fully dispersive model equation is derived which describes the propagation of waves in a thin elastic body overlying an incompressible inviscid fluid. The equation is nonlocal in the linear part, and is similar to the so-called Whitham equation which was proposed as a model for the description of wave motion at the free surface of an inviscid fluid.Steady solutions of the fully nonlinear hydro-elastic Euler equations are approximated numerically, and compared to numerical approximations to steady solutions of the fully dispersive but weakly nonlinear model equation.The bifurcation curves for these two different models are compared, and it is found that the weakly nonlinear model gives accurate predictions for waves of small to moderate amplitude. For larger amplitude waves, the two models still agree on key qualitative features such as the bifurcation points, secondary bifurcations, and the number of oscillations in a given fundamental wave period.

Unstable Slip Pulses and Earthquake Nucleation as a Nonequilibrium First-Order Phase Transition.

The onset of rapid slip along initially quiescent frictional interfaces, the process of "earthquake nucleation," and dissipative spatiotemporal slippage dynamics play important roles in a broad range of physical systems. Here we first show that interfaces described by generic friction laws feature stress-dependent steady-state slip pulse solutions, which are unstable in the quasi-1D approximation of thin elastic bodies. We propose that such unstable slip pulses of linear size L^{*} and characteristic amplitude are "critical nuclei" for rapid slip in a nonequilibrium analogy to equilibrium first-order phase transitions and quantitatively support this idea by dynamical calculations. We then perform 2D numerical calculations that indicate that the nucleation length L^{*} exists also in 2D and that the existence of a fracture mechanics Griffith-like length L_{G}<L^{*} gives rise to a richer phase diagram that features also sustained slip pulses.

A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

A 2D BEM–FEM approach for time harmonic fluid–structure interaction analysis of thin elastic bodies

Abstract This paper deals with two-dimensional time harmonic fluid–structure interaction problems when the fluid is at rest, and the elastic bodies have small thicknesses. A BEM–FEM numerical approach is used, where the BEM is applied to the fluid, and the structural FEM is applied to the thin elastic bodies. From the fluid point of view, the thin elastic bodies are considered of null thickness. This assumption is treated using simultaneously the Singular Boundary Integral Equation and the Hypersingular Boundary Integral Equation. It is assumed that the thin elastic bodies are under the Euler–Bernoulli hypotheses with added rotational inertia. The BEM equations (fluid) and the FEM equations (thin bodies) are coupled using appropriate equilibrium and compatibility conditions. The developed BEM–FEM model requires a simple discretization and leads to a small number of degrees of freedom, although it has some limitations that are studied in some depth. This approach is validated with existing results in the field of sound barriers, and new results using complex barrier shapes are presented. Also, a parametric study about a straight wall immersed in a fluid is done, which provides results of practical usage.

Geometry and fragmentation of soft brittle impacted bodies

Some mechanisms of shape selection and fragmentation patterns of impacted or strongly solicited elastic bodies of simple geometry are reviewed. By strongly solicited we mean objects whose shape undergoes order one deformations, a situation typically encountered with soft bodies, possibly leading to fracture. We are particularly interested in the development of elastic instabilities and their relevance to interpret fragmentation experiments. We review how the shape of elastic slender objects like rods and membranes is affected by buckling instabilities when impacted, and how brittle sheets break through a well-defined pattern of cracks. Central to the understanding of this pattern is the concept of optimal fragmentation. The crack pattern of thin elastic bodies, including the diverging concertina tearing, the perforation of a thin sheet by a cone and the star shape pattern of impacted windows are optimal objects, ruled by a balance between stored elastic energy and surface creation energy.

On the Samarskii-Andreev transmission conditions in the theory of elastic beams

It is proved that the transmission conditions for elastic beams in the case of a nonideal joint are limiting in the construction of asymptotics for the transmission problem for two thin elastic bodies if the boundary between the bodies is filled by slightly extendible material.

Growth and shape control of disks by bending and extension

Differential growth of thin elastic bodies furnishes a surprisingly simple explanation of the complex and intriguing shapes of many biological systems, such as plant leaves and organs. Similarly, inelastic strains induced by thermal effects or active materials in layered plates are extensively used to control the curvature of thin engineering structures. Such behaviour inspires us to distinguish and to compare two possible modes of differential growth not normally compared to each other, in order to reveal the full range of out-of-plane shapes of an initially flat disk. The first growth mode, frequently employed by engineers, is characterised by direct bending strains through the thickness, and the second mode, mainly apparent in biological systems, is driven by extensional strains of the middle surface. When each mode is considered separately, it is shown that buckling is common to both modes, leading to bistable shapes: growth from bending strains results in a double-curvature limit at buckling, followed by almost developable deformation in which the Gaussian curvature at buckling is conserved; during extensional growth, out-of-plane distortions occur only when the buckling condition is reached, and the Gaussian curvature continues to increase. When both growth modes are present, it is shown that, generally, larger displacements are obtained under in-plane growth when the disk is relatively thick and growth strains are small, and vice versa. It is also shown that shapes can be mono-, bi-, tri- or neutrally stable, depending on the growth strain levels and the material properties: furthermore, it is shown that certain combinations of growth modes result in a free, or natural, response in which the doubly curved shape of disk exactly matches the imposed strains. Such diverse behaviour, in general, may help to realise more effective actuation schemes for engineering structures.

The decomposed form of the stress state for transversely isotropic beam bending based on elastic theory

AbstractIn this paper we study a plane stress problem of rectangular beams which are slender and thin elastic bodies with free upper and lower surfaces and in the absence of body forces. Without employing ad hoc stress assumptions, the decomposed form of the stress state for transversely isotropic beam bending is proposed on the basis of the classical elastic theory, and the corresponding decomposition theorem is inextenso presented for the first time. It is shown that the stress state of transversely isotropic beams with free faces can be uniquely decomposed into two parts: the interior state and the Papkovich‐Fadle (P‐F) state. In the proof course of the decomposition theorem, some basic mathematical methods are used only, so the proof is more convenient for being understood. These two states are identical to two equations derived in the refined theory of transversely isotropic beams, so this work practically proves the consistency between the decomposed form and the refined theory of transversely isotropic beams. Moreover, with the separate consideration of the interior state and P‐F state, a relatively simple analytical solution is often practical and desirable, and the numerical computation process is greatly simplified.

Finding minimum energy configurations for constrained beam buckling problems using the Viterbi algorithm

In this work, we present a novel technique to find approximate minimum energy configurations for thin elastic bodies using an instance of dynamic programming called the Viterbi algorithm. This method can be used to find approximate solutions for large deformation constrained buckling problems as well as problems where the strain energy function is non-convex. The approach does not require any gradient computations and could be considered a direct search method. The key idea is to consider a discretized version of the set of all possible configurations and use a computationally efficient search technique to find the minimum energy configuration. We illustrate the application of this method to a laterally constrained beam buckling problem where the presence of unilateral constraints together with the non-convexity of the energy function poses challenges for conventional schemes. The method can also be used as a means for generating “very good” starting points for other conventional gradient search algorithms. These uses, along with comparisons with a direct application of a gradient search and simulated annealing, are demonstrated in this work.

Optimal design of V-type ultrasonic motor

V-type ultrasonic linear motor fabricated using a simple punching technique was proposed to utilize as an actuator of small precision machine. The stator of the motor is composed of a thin elastic body and four ceramics attached to the upper and bottom areas of the body. The ceramics have each direction of polarization. When two harmonic voltages with a 90° phase difference are applied to the ceramics, symmetric and anti-symmetric displacements will generate at the tip to produce an elliptical motion. A finite element analysis (ATILA) was conducted to simulate the motion pattern for the contact tip of the stator. To develop a model that generates the maximum displacement at contact tip, the FEM program was used for various lengths. In addition, an optimal model was chosen by considering the magnitude and shape of the displacement according to changes in frequency. The maximum elliptical displacement is shown by W2L11 model, which has a ratio of ceramic width to length of 1:5.5. However, the displacement of the contact tip is reduced by the bucking phenomenon if the ratio is larger than 1:6.

Buckling transition and boundary layer in non-Euclidean plates

Non-Euclidean plates are thin elastic bodies having no stress-free configuration, hence exhibiting residual stresses in the absence of external constraints. These bodies are endowed with a three-dimensional reference metric, which may not necessarily be immersible in physical space. Here, based on a recently developed theory for such bodies, we characterize the transition from flat to buckled equilibrium configurations at a critical value of the plate thickness. Depending on the reference metric, the buckling transition may be either continuous or discontinuous. In the infinitely thin plate limit, under the assumption that a limiting configuration exists, we show that the limit is a configuration that minimizes the bending content, among all configurations with zero stretching content (isometric immersions of the midsurface). For small but finite plate thickness, we show the formation of a boundary layer, whose size scales with the square root of the plate thickness and whose shape is determined by a balance between stretching and bending energies.

Stability of Slender Bodies under Compression and Validity of the von Kármán Theory

Since their formulation almost 100 years ago, the von Kármán (vK) plate equations have been frequently used both by engineers and by analysts to study thin elastic bodies, in particular their stability behaviour under applied loads. At the same time the derivation of these equations met some harsh criticism and their precise mathematical status has been unclear until very recently. Following up on a recent variational derivation of the vK theory by Friesecke, James and Müller from three-dimensional nonlinear elasticity we study the predictions and the validity of the vK equation in the presence of in-plane compressive forces. The first main result is a stability alternative: either the load leads to instability already in the nonlinear bending theory of plates (Kirchhoff–Love theory), or it leads to an instability in a geometrically linear KL theory (‘linearized instability’), or vK theory is valid. The second main result states that under suitable conditions the critical loads for nonlinear stability and linearized instability coincide. The third main result asserts this critical load also agrees with the load beyond which the infimum of the vK functional is −∞. The main ingredients are a sharp rigidity estimate for maps with low elastic energy and a study of the properties of isometric immersions from a set in \({\mathbb{R}^2}\) to \({\mathbb{R}^3}\) and their geometrically linear counterparts.

Convergence of Equilibria of Thin Elastic Plates – The Von Kármán Case

We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch 4 (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers.

Thin-plate theory for large elastic deformations

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff–Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.

3D Wave Field Scattered By Thin Elastic BodiesBuried In An Elastic Medium UsingThe Traction Boundary Element Method

A combined formulation based on the Traction Boundary Element Method (TBEM) and the Boundary Element Method (BEM) is proposed to model three-dimensional wave scattering in solid media, which contain thin elastic bodies whose geometry remains the same along one direction. Although the classical BEM models degenerate in the presence of thin heterogeneities, this combined formulation, in which one side of the body is discretized with the BEM and the other side with the TBEM formulation, is able to model the wave field in the vicinity of these heterogeneities. Arbitrary-shaped and oriented elastic thin bodies, even with no thickness, may be modelled using this formulation. Analytical integrations are performed to evaluate the hypersingular integrals. The proposed model is applied to two different thin heterogeneities buried in an elastic unbounded medium.

A priori modeling error estimates of hierarchical models for elasticity problems for plate- and shell-like structures

Here a hierarchical model for thin elastic bodies is presented in which the variations of the displacement field through the thickness can be represented as a sequence of apolynomials. An a priori hierarchical modeling error estimate is derived and it is found that the modeling error decreases monotonically as the order of the thickness-polynomials increases and has a uniform convergence rate with respect to the thickness. Numerical results are given illustrating the theoretical results.

Lax–Phillips theory for scattering by thin elastic bodies

The scheme of construction of linear self-adjoint operators for acoustic systems with an interaction of sound and vibration is proposed. Kinetic energy of a system induces the basic Hilbert space L2(Ω), where Ω denotes the domain filled with fluid. Potential energy of the system induces the quadratic form on this Hilbert space and, consequently, the positive operator that governs the evolution. Unlike earlier theories, our operators are perturbations of the Laplace operator in L2 spaces. The operators may also be obtained as extensions of symmetric operator (−Δ) defined on C∞0(Ω) into a wider Hilbert space. Functions from their domains have an additional smoothness of their derivatives on the boundary; the smoothness follows from the elasticity of the boundary. The scheme allows one to describe inhomogeneities of vibrating objects concentrated at a point or on a line (such as cracks and stiffeners). It is demonstrated that Lax–Phillips scattering theory may be generalized for scattering by thin elastic objects. The necessary condition for the existence of the incoming and the outgoing subspaces is derived in terms of spectral properties of the operator of the isolated elastic object.

Internal and boundary calculations of thin elastic bodies

A method for the separate construction of the main stress-strain state (the internal calculation) and the boundary corrections (the boundary calculations) are discussed in the case of a linear static problem in the theory of shells and plates. It is assumed that the internal calculation is carried out using an iterative process based on the Kirchhoff-Love theory. The boundary calculation involves the construction of antiplane and plane boundary layers, that is, in the initial approximation they reduce to the solution of antiplane and plane problems in the theory of elasticity. Investigation of the asymptotic behaviour of the boundary corrections shows that near a weakly clamped edge only the correction from the antiplane boundary layer is important and that near a fairly rigidly clamped edge only the correction from the plane boundary layer is important. The advisability of the use of the shear theory of the bending of plates for investigating boundary elastic phenomena is discussed from the point of view of the results obtained. It is shown that, close to the free edge, its use is justified and is adequate for the method described in the paper both with regard to the numerical results and with regard to the nature of the mathematical apparatus. As a method for investigating boundary elastic phenomena, shear theories lose their meaning close to a fairly rigidly clamped edge since they only enable one to construct the minor part of the correction asymptotically.