The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

Linear Equations of Motion in Finite Elasticity

In this note we show that it may be possible and useful to construct valid strain-energy functions that lead directly to linear equilibrium equations for problems in isotropic homogeneous unconstrained nonlinear elasticity. While it is possible to make some general progress the final outcome will depend on the geometry and kinematics of the problem under consideration. Specific examples are given to show how exact solutions, via the linear equations of motion, can be found to non-trivial problems for physically meaningful constitutive models.      

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Valanis-Landel Strain-Energy Function

A constitutive equation is derived for the Cauchy stress matrix for arbitrary deformations of an isotropic elastic solid characterized by a Valanis-Landel strain-energy function. A simple example is given of the way in which results for controllable deformations of an incompressible elastic solid, with a Valanis-Landel strain-energy function, can be obtained from the known results for the more general strain-energy function employed by Rivlin.     

Generalisations of the strain-energy function of linear elasticity to model biological soft tissue

Strain measures consistent with the linear, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, and non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue.     

Finite Third-order Gradient Elasticity and Thermoelasticity

A constitutive format for the third-order gradient elasticity is suggested. It includes both isotropic and anisotropic non-linear behavior under finite deformations. Appropriate invariant stress and strain variables are introduced, which allow for reduced forms of the elastic energy law that identically fulfill the objectivity requirement. After working out the transformation behavior under a change of the reference placement, the symmetry transformations for third-order materials can be introduced. After the mechanical third-order theory, an extension to thermoelasticity is given, and necessary and sufficient conditions are derived from the Clausius-Duhem inequality.     

Elastic solids with different moduli in tension and compression

A new approach is given to the theory of non-linear elastic materials which have different behaviour in tension and compression. Two applications are made to incompressible non-linear materials using general forms for the strain energy functions. The linear form of the theory is shown to be equivalent to that used by previous writers.     

Extension,inflation and torsion of a residually stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.     

Bifurcation and stability of homogeneous deformations of an elastic body under dead load tractions with Z 2 symmetry

An analysis is given of bifurcation and stability of homogeneous deformations of a homogeneous, isotropic, incompressible elastic body subject to three perpendicular sets of dead-load surface tractions of which two have equal magnitude. A minimization problem is formulated within the framework of non-linear elasticity, which leads to a bifurcation problem with Z ₂ symmetry. Various bifurcation diagrams are deduced by using singularity theory, and stabilities of solution branches are examined.     

Gradient-based non-linear microstructure design

Fourier analysis is implemented on the orientation distribution of a polycrystalline microstructure. The linearity and convexity of the Fourier space, with respect to orientation, allows one to consider all possible distributions by considering all linear combinations of single-grain orientations. The limits of the Fourier space are therefore defined by the solutions to a set of linear programming problems. A unique approach to the linear programming, similar to the Krylov subspace methods for obtaining solutions to linear systems, is presented. The method is particularly efficient for this application where a large number of independent variables is often required. These solutions are then used as the constraints in the gradient-based optimization of non-linear functions within the Fourier space. In the example, Taylor yield theory and an anisotropic solution for the stress concentration around a hole in a plate of cubic-orthotropic polycrystalline material are expressed as non-linear functions within the Fourier space. The maximum obtainable ratio of Taylor factor to stress concentration for any polycrystalline orientation distribution in copper is found to be 1.22, more than double the minimum value.     

Towards the solution of finite plane-strain problems for compressible elastic solids

A new approach to the solution of finite plane-strain problems for compressible Isotropie elastic solids is considered. The general problem is formulated in terms of a pair of deformation invariants different from those normally used, enabling the components of (nominal) stress to be expressed in terms of four functions, two of which are rotations associated with the deformation. Moreover, the inverse constitutive law can be written in a simple form involving the same two rotations, and this allows the problem to be formulated in a dual fashion.For particular choices of strain-energy function of the elastic material solutions are found in which the governing differential equations partially decouple, and the theory is then illustrated by simple examples. It is also shown how this part of the analysis is related to the work of F. John on harmonic materials.Detailed consideration is given to the problem of a circular cylindrical annulus whose inner surface is fixed and whose outer surface is subjected to a circular shear stress. We note, in particular, that material circles concentric with the annulus and near its surface decrease in radius whatever the form of constitutive law within the given class. Whether the volume of the material constituting the annulus increases or decreases depends on the form of law and the magnitude of the applied shear stress.     

Hypoplasticity theory for granular materials—I: Two-dimensional plane strain exact solutions

Numerous theories have been developed in order to gain a better understanding of the behaviour of granular materials. One such theory, originally developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe over the last decade or so, is the rational continuum mechanical theory termed hypoplasticity. This theory involves a constitutive law for which the stress-rate is a properly invariant isotropic tensorial function of the stress- and strain-rate tensors, but possesses a non-differentiable dependence on the strain-rate tensor. From a practical perspective, it would be highly desirable to determine simple solutions of hypoplasticity applying to a range of fundamental problems, such as gravity flow in a two-dimensional wedge-shaped hopper. Although this is the original motivation of this study, the complexity of the theory appears to preclude the determination of simple analytical solutions, such as the classical solution of Jenike applying to the Coulomb-Mohr granular solid. In this paper, we undertake a mathematical investigation to determine solutions for two-dimensional steady quasi-static plane strain compressible gravity flow for hypoplastic granular materials. For certain special cases we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solution is not a trivial matter. Three-dimensional axially symmetric solutions analogous to those given in this present study are presented in a companion paper, part II.     

The Attainable Region of strain-invariant space for elastic materials

Within the theory of isothermal isotropic non-linear elasticity, the selection of the appropriate form for the strain energy function W in terms of the strain invariants is still an issue. The purpose of this paper is to introduce ideas and techniques which it is hoped will contribute to the task of finding an appropriate form for the strain energy. Three principal ideas are developed in this paper. Firstly, not all of invariant-space corresponds to real deformations. Constitutive equations only need to match real behaviour over a restricted part of invariant space, called the Attainable Region, bounded by states of deformation corresponding to uniaxial and equi-biaxial extension. Secondly, examples are given of how to exploit the fact that the Attainable Region is restricted. Mapping a deformation onto this region allows visualization of how close the deformation is to the well-understood uniaxial, equi-biaxial and simple shear deformations, and how this varies in space or time. Thirdly, acceptable strain invariants do not have to be obviously symmetric functions of the principal stretches. The ordered principal stretches are themselves invariants, and explicit unique algebraic expressions can be given through which the greatest, middle or least stretch can be calculated in terms of the usual invariants. Thus invariants can be chosen which are apparently non-symmetric functions of the ordered stretches.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

Axisymmetric Thermoelastoplastic Stress State of Isotropic Solids of Revolution Under Impulsive Loading

Consideration is given to the solution of a dynamic problem for a solid of revolution with an arbitrary meridional section under impulsive thermomechanical loading inducing elastoplastic strains. The theory of small elastoplastic deformations is used. The constitutive equations are linearized by the variable-parameter method. The unloading process is described by a linear law. The solution technique involves the finite-element approximation in spatial coordinates and the finite-difference representation of time derivatives. Based on the principle of linear summation, recurrent relations are derived for successive evaluation of nodal displacements by an explicit scheme in time. Solution for cylinders and disks are presented to illustrate the influence of elastoplastic deformations on wave processes     

A new type of non-linear approximation with application to the duffing equation

A new type of trial solution which differs from the usual linear combination of approximating functions is considered. It involves modifying the approximating functions with “form functions;” functions containing undetermined parameters appearing non-linearly, the proper choice of which provide a closer approximation to the large local curvatures which appear in some non-linear problems. In this paper the “form function” approximation is demonstrated for steady-state solutions of the Duffing equation. This equation arises in the problem of non-linear vibration of buckled beams and plates. It is shown that the stability behavior of these steady-state solutions is governed by a Hill equation. It is found that the “form function” approximation gives noticeably better numerical results than, for example, those given by the harmonic balance method. The method also provides additional insight into the non-linear behavior, particularly in the low frequency response region.     

Self-similar problems of elastic contact for non-convex punches

Self-similar problems of contact for non-convex punches are considered. The non-convexity of the punch shapes introduces differences from the traditional self-similar contact problems when punch profiles are convex and their shapes are described by homogeneous functions. First, three-dimensional Hertz type contact problems are considered for non-convex punches whose shapes are described by parametric-homogeneous functions. Examples of such functions are numerous including both fractal Weierstrass type functions and smooth log-periodic sine functions. It is shown that the region of contact in the problems is discrete and the solutions obey a non-classical self-similar law. Then the solution to a particular case of the contact problem for an isotropic linear elastic half-space when the surface roughness is described by a log-periodic function, is studied numerically, i.e. the contact problem for rough punches is studied as a Hertz type contact problem without employing additional assumptions of the multi-asperity approach. To obtain the solution, the method of non-linear boundary integral equations is developed. The problem is solved only on the fundamental domain for the parameter of self-similarity because solutions for other values of the parameter can be obtained by renormalization of this solution. It is shown that the problem has some features of chaotic systems, namely the global character of the solution is independent of fine distinctions between parametric-homogeneous functions describing roughness, while the stress field of the problem is sensitive to small perturbations of the punch shape.     

Geometrical methods in the analysis of ordinary differential equations

Summary In several fields of engineering research, particularly in the study of vibrations, electrical circuits and in some problems of fluid mechanics, approximations which lead to linear differential equations are proving inadequate. This circumstance is focussing the attention of research workers and engineers on non-linear problems. This article gives an account, without proofs, but with literature references, of methods for the qualitative integration of non-linear ordinary differential equations of the first order, i.e. for the determination of the pattern of the integral curves of such equations. The use of such geometrical methods becomes necessary in cases when the equation cannot be integrated in closed form. Simple and complex patterns associated with singular points are discussed, and criteria for their classification are given. A method of determining the asymptotic behaviour of the family of solutions is given, and criteria for the existence of closed curves in the family of solutions, as well as the occurrence of limit cycles, are discussed. A brief discussion of the Kronecker index and of the mutual relation between several singular points is added. The text is illustrated with several examples selected from the fields of vibration, compressible fluid flow and electrical circuits.     

An asymptotic analysis of finite deformations near the tip of an interface-crack

This work investigates the behavior of a traction-free crack at the interface of two semi-infinite slabs bonded together under the conditions of plane strain. A determination of the mathematical form of the deformation and stresses near the crack-tip, consistent with the fully non-linear equilibrium theory of compressible elastic solids, is found by an asymptotic treatment of the deformation.Each slab is assumed to be hyperelastic, homogeneous, and isotropic with Knowles-Sternberg type asymptotic conditions on its strain-energy density. It is shown that under these conditions, the interface-crack problem admits solutions in which oscillatory singularities do not occur. This suggests that it is the approximations made by the linear theory which produce these singularities.