The influence of the type of loading on the asymptotic behavior of slender elastic rings

We study the influence of the type of loading on the asymptotic behavior of linearly elastic, isotropic and homogeneous slender circular rings. By using formal asymptotic expansions, we obtain three families of models depending on the properties of the loads. If the loads expend work in inextensional displacements, then we find the classical model where the leading term of the energy corresponds to the bending-torsion energy of inextensional displacements. If the loads do no work in inextensional displacements, the model must be refined and we obtain two other types of models. In these other models, which depend on the type of loading, the leading term of the energy contains additional terms such as, for the second class, an extension energy due to the circumferential stretching of the ring, and even, for the third class, specific load-dependent contributions. This classification is illustrated in several examples.     

A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼h^β, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼h^α. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)³→ℝ³, the L² distance of ∇v from a single rotation is bounded by a multiple of the L² distance from the set SO(3) of all rotations.     

Bending Boundary Layers in a Moving Strip

The high-speed transport of thin-sheet materials occurs in avariety of industrial processes. In this paper the method of matchedasymptotic expansions is used to solve the low tension, large-deflectionshape of a convected strip hanging under gravity. To leading order theequations for the deflection in the boundary layer and in the centre ofthe span are nonlinear. Previous treatments of this problem have notaccurately solved the leading-order equation within the boundary layer.The composite analytic solution of the full nonlinear equation isobtained and compares well with numerical solutions.     

Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.     

Energy Minimising Properties of the Radial Cavitation Solution in Incompressible Nonlinear Elasticity

Consider an incompressible, hyperelastic material occupying the unit ball B⊂ℝ ⁿ in its reference state. Suppose that the deformation u:B→ℝ ⁿ is specified on the boundary by

On the Optimal Location of Singularities Arising in Variational Problems of Nonlinear Elasticity

Experiments on elastomers have shown that triaxial tension can induce a material to exhibit holes that were not previously evident. Analytic work in nonlinear elasticity has established that such cavity formation may indeed be an elastic phenomenon: sufficiently large prescribed boundary deformations yield a hole-creating deformation as the energy minimizer whenever the elastic energy is of slow growth. One of the many unanswered problems is where such holes will form. In this paper we suggest a new method, which is based upon asymptotics and linear elasticity, that can be used to determine the optimal location for hole creation. Using this method we show that, under reasonable hypotheses, the center is (locally) the best position for a solitary hole to form in an elastic ball. This revised version was published online in August 2006 with corrections to the Cover Date.     

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

Abstract

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.     

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.     

A Circular Inclusion with Inhomogeneously Imperfect Interface in Plane Elasticity

A general method is presented for the rigorous solution of a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is considered to be imperfect with the assumption that the interface imperfections are circumferentially inhomogeneous. Using analytic continuation, the basic boundary value problem for four analytic functions is reduced to two coupled first order differential equations for two analytic functions. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity and certain other auxiliary conditions. The method is illustrated using a particular class of inhomogeneous interface. The results from these calculations are compared to the corresponding results when the imperfections in the interface are circumferentially homogeneous. These comparisons illustrate, for the first time, how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the average stresses induced within the inclusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

用有限元广义混合法分析不可压缩或几乎不可压缩弹性体

不可压缩或几乎不可压缩问题在数学上表现为最小 势能原理中的某些项趋于无穷大,使得有限元方程产生病态。本文给出了不可压缩或几乎不可压缩弹性分析的广义混合变分原理,以此为基础建立了该类问题的有限元广义混合法。该变分原理的泛函中不含有上面这种奇异项,故其有限元方程不会产生病态。算例表明该有限元法可以同时进行可压缩、不可压缩或几乎不可压缩弹性分析,且精度良好;有限元常规位移法及Hermann法是该法的特例。     

Weak Local Minimizers in Finite Elasticity

This paper deals with necessary conditions and sufficient conditions for a weak local minimum of the energy of a hyperelastic body. We consider anisotropic bodies of arbitrary shape, subject to prescribed displacements on a given portion of the boundary. As an example, we consider the uniaxial stretching of a cylinder, in the two cases of compressible and incompressible material. In both cases we find that there is a continuous path across the natural state, made of local energy minimizers. For the Blatz-Ko compressible material and for the Mooney-Rivlin incompressible material, explicit estimates of the minimizing path are given and compared with those available in the literature. Dedicated to the memory of Victor J. Mizel.     

非线性弹性及全量弹塑性有限元分析中的一致性切线模量

本文导出了非线性弹性及全量弹塑性有限元分析中的一致性切线模量,从而可以保持牛顿迭代法固有的平方收敛速度.指出了某些文献中关于切线模量的不正确表述,并以数值算例验证了本文方法的正确有效.     

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.     

Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

The Lagrange Multiplier in Incompressible Elasticity Theory

The first variation condition for the potential energy in nonlinear elasticity for incompressible materials provides a linear functional which vanishes on an appropriately constrained set of variations. We prove a representation theorem for such linear functionals which forms the basis for the existence of a constraint reaction (Lagrange multiplier) field. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equations of Motion

Slender thread like bodies (like cables, ropes, textilethreads or belts) are often used in technical applications. Becauseof their dimensions the one-dimensional continuum is the appropriatemechanical model for bodies of this type. Making use of the basicrelations of three-dimensional continua as a starting point the paperdevelops the general kinematic and kinetic relations of one-dimensionalcontinua for the case that the cross-sections will remain plane (Bernoullihypothesis), that large deflections are possible but the strains remainsmall and that the material is homogeneous and isotropic and behaveslinearly elastic. This results in the equations of motion of shearableand extensible rods (Timoshenko-beams). By neglection of shear deformationand of the rotational inertia of the cross-sections (assumptions thatcan be done in most technical applications) the equations of motionof Euler–Bernoulli-beams are derived in standard and concentratedform. The Euler–Bernoulli-beam equations contain the equations ofmotion of threads with zero bending and torsional stiffness. It isshown that the neglection of bending and torsional stiffness is onlyvalid if the tension is always positive. The second part of this paper[1] selects and develops appropriate numerical solution methods.The derived algorithms are used to solve problems from space and marineengineering.     

A Frame-Free Formulation of Elasticity

As I pointed out at the end of Sect. 4 in [6] of my booklet Five Contributions to Natural Philosophy, it should be possible to make the principle of material frame-indifference vacuously satisfied by using an intrinsic mathematical frame-work that does not use an external frame-space at all when describing the internal interactions of a physical system. Here I will do just that for the classical theory of elasticity and also for the theory of hyperelasticity, i.e., elasticity based on a strain-energy function. I will also comment on possible restrictions on the corresponding intrinsic response functions.This paper is based, in part, on lectures that I gave on June 29, 2005 at the meeting in Reggio-Calabria in honor of the 65th birthday of Gianpietro Del Piero and on July 6, 2005, at the University of Messina.     

The Cauchy Relations in Linear Elasticity Theory

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The vanishing of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.     

Plane Deformations in Incompressible Nonlinear Elasticity

The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Azimuthal Shear in Compressible Finite Elasticity

In an earlier paper, the broadest classes of compressible isotropic strain energies that support irrotational universal deformations were identified and the problems of cylindrical and spherical inflation or compaction were solved in closed form for all of these strain energies. Similar closed form solutions of the problem of azimuthal shear are presented here.