A Theory of Non-Linear Elasticity Compatible With the Murnaghan Equation of State

We present an equation of state for a cubic non-linear elastic material in a general state of finite strain. For hydrostatic pressure, the predictions closely follow Murnaghan's well-known equation of state. At 170 kbar, our model differs from Murnaghan's equation by only 1.3%, which contrasts with the currently accepted non-linear elasticity theory that differs by 10% at this pressure. The theory is based on expressing the variation of the elastic constants as a linear function of stress rather than strain. We define a different set of third-order elastic constants, which involve a derivative with respect to stress, and relate these to the conventional third-order elastic constants. We apply the model to GaAs under hydrostatic pressure and we compare the predictions of the conventional non-linear theory with those of the model we present.     

Exact and approximate solutions of Riemann problems in non-linear elasticity

Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to non-linear elastic theory. In this paper one such class of solver is examined based upon characteristic tracing in conjunction with high-order monotonicity preserving weighted essentially non-oscillatory (MPWENO) reconstruction. Furthermore, a new iterative method for finding exact solutions of the Riemann problem in non-linear elasticity is presented. Access to exact solutions enables an assessment of the performance of the numerical techniques with focus on the resolution of the seven wave structure. The governing model represents a special case of a more general theory describing additional physics such as material plasticity. The numerical scheme therefore provides a firm basis for extension to simulate more complex physical phenomena. Comparison of exact and numerical solutions of one-dimensional initial values problems involving three-dimensional deformations is presented.     

Piezoelectricity of chiral nematic elastomers

A molecular model of freely jointed chains of chiral monomers is developed to describe the piezoelectric effect in chiral nematic elastomers. The model, an extension of the neo-classical theory of nematic polymer networks, takes into account a chiral biasing of molecular alignment under shear which leads to induced polarisation if the monomers contain a transverse dipole moment. The resulting theory is fully non-linear in elastic deformations, in the spirit of ordinary rubber elasticity. The expansion to the highest order in small strains gives the three linear piezoelectric coefficients predicted by phenomenological models. Received 7 September 1998 and Received in final form 19 October 1998     

Vibrations of a system with memory,non-linear elasticity,friction and relaxation

The solution of a third-order non-linear differential equation with slowly varying coefficients and small time lag is found. This equation governs processes with significant damping, and an application is made to a mechanical vibrating system with non-linear elasticity, internal friction and relaxation.     

On the berger approximation: A critical re-examination

Since the first paper by Berger some two decades ago, the simplification known as the Berger approximation has been invoked by the authors of several score papers in spite of the fact that no rational mechanical basis for the approximation could be found. Many recent papers have raised doubts on its applicability. In this paper, using certain well known results from the two dimensional theory of elasticity, a plausible explanation for the origin of the Berger method is suggested. The arguments can be developed further to show that other specious Berger-like approximations can be developed, all of them leading to uncoupled non-linear equations yielding different overall results. Further, it is shown that such methods fail to predict the non-linear behaviour with respect to important parameters and that whatever accuracy is obtained in the solution of a particular problem can at best be attributed to fortuity.     

Buckling of a compressed elastic membrane: a simple model

The buckling of a folded membrane submitted to a bi-axial compression is studied in the framework of the continuum non-linear elasticity theory. We show that the formation of the fold patterning can be quantitatively well described with a simple non-linear model. As a matter of fact, with this model, we recover the experimental phase diagram of a secondary buckling instability with a very good precision. In addition, depending on the anisotropy of the applied compressive stress, we find that the buckling coarsening dynamics can be described as a 1D spinodal decomposition (for a uni-axial stress) or as a 2D XY model (for an isotropic bi-axial stress) with an irrotational non-scalar order parameter. For an isotropic bi-axial stress, we indeed recover the famous coarsening exponent: n=1/4. This exponent has to be confirmed experimentally.     

Evaluation of eigenfunctions from compound matrix variables in non-linear elasticity – I. Fourth order systems

We show how the compound matrix method can be used to produce eigenfunctions as well as eigenvalues for bifurcation problems in non-linear elasticity. For typical problems in elasticity the boundary conditions require a different treatment to that required for typical problems in fluid mechanics. For elasticity problems we have to use an additional shooting method to ensure that the boundary conditions are satisfied.     

THE DYNAMICS OF A VIBROMACHINE WITH PARAMETRIC EXCITATION

The dynamics of a vibration machine with piecewise linear elastic ties under parametric harmonic excitation is investigated. Different designs of elastic elements with periodically time-varying elasticity are described. Specific non-linear features of parametric oscillations in the system under study are revealed (the invariance of parametric vibration regime to possible disturbance of phase co-ordinates, conditions of limitedness of amplitude of parametric vibrations, spectral features of non-linear parametric regimes, etc.). By the utilization of these non-linear effects, a procedure for the design of the main parameters of a parametric vibromachine is proposed.     

Bending,vibration and buckling of simply supported thick multilayered orthotropic plates: An evaluation of a new displacement model

Based upon a piecewise linear displacement field which allows the contact conditions for the displacements and the transverse shearing stresses at the interfaces to be satisfied simultaneously, the non-linear (in the von Kármán sense) equations of motion for thick multilayered orthotropic plates are developed. Successively, the equations are specified to the linear boundary value problem of the bending and to the linear eigenvalue problems of the undamped vibration and buckling of rectangular plates. In order to assess the accuracy of the proposed theory, the sample problem of the bending, free undamped vibration and buckling of a three-layered, symmetric cross-ply, square plate simply supported on all edges is investigated. For purposes of comparison, numerical results from the exact elasticity theory, the classical lamination (Kirchhoff) theory and the shear deformation theory (Timoshenko and Mindlin) with three different values of the shear correction factor are also presented. It is found that the proposed approach is very efficient in predicting the global responses (deflection, natural frequencies and buckling loads) of thick multilayered plates and models effects, such as the distortion of the deformed normals, not attainable from the classical lamination theory, as well as the shear deformation theory.     

Conjugaisons finies et milieux continus en relativité générale

We present here the main elements of a mechanics of relativistic continua relying upon a concept of 《finite conjugacy》 between two relativistic motions described by two unit vector-fields u and u' defined on two different relativistic manifolds $\text{M}$-and $\text{M}$'.This purely relativistic, global, and intrinsic theory leads, together with a new approach of the deformation tensors in relativity, to a differential system of equations for the conjugacies which is neither under-determined nor over-determined. A rough study of the propagation of the conjugacy-waves shows then that it is advisable to consider the notion of a finite conjugacy as a satisfying relativistic extension of the classical and tridimensional notion of a finite deformation in mechanics, and to identify the spatial conjugacy-waves obtained with the ordinary acoustic waves.Drastic particularizations of the space-times $\text{M}$ and $\text{M}$', of the motions u and u', of the admissible types of conjugacies and of the elastic behaviour of the continua under study allow to recover, as very important but particular cases, the tridimensional non-relativistic theory of elasticity for finite deformations and non-linear behaviour, as well as the main theories of relativistic elasticity already proposed by Mmes Choquet-Bruhat and Lamoureux-Brousse, Rayner, Carter and Carter-Quintana, Grot-Eringen…. The obtained system of equations generalizes also to the finite case some aspects of the infinitesimal theory of Weber and Papapetrou.     

Some New Features of Linear Theory for Describing Non-linear Diffusion

The non-linear flux equation, the non-linear Fokker-Planck equation (or Smoluchowski equation), and the non-linear Langiven equation are the basicequations for describing particle diffusion in non-ideal system subjected totime-dependent external fields. Nevertheless, the exact solution of thoseequations is still a challenge because of their inherent complexity of thenon-linear mathematics. Li et al. found that, based on the defined apparentvariables, the non-linear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or local equilibrium assumption. In this paper, some new features of the theory were found: (i) The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; (ii) The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planck equation, can be derived more consistently from the microscopic molecular statistical theory; (iii) In the theory, thenon-linear Langiven equation also bears a linear form; (iv) For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.     

The variational and virial-like theory of oscillations and stability of non-conservative and/or non-linear mechanical systems

This paper presents the general theory and some applications of Hamiltonian action and virial-like methods to the exact and/or approximate study of the periodic solutions of non-conservative and/or non-linear, but holonomic, oscillators. Specifically, the first (theoretical) part covers: (i) a generalization of “Hamilton's law of varying action”, to include variable time-endpoints (i.e., frequency variations) and variable system parameters, such as elasticity and/or inertia, and (ii) a general formulation of the “virial” theorem and its use in determining the stability/instability of given oscillatory motions. Applications of the above (of the Rayleigh-Ritz type) to the following systems are then presented: (i) linear circulatory; (ii) linear conservative, loading parameter dependent, (iii) Duffing's (cubic) oscillator; (iv) van der Pol's oscillator, and related general non-linear and non-conservative case. Comparison of the method with other existing ones, and related open problems (such as limit-cycle stability), are finally discussed.     

On a linearization program of non-linear field equations

The relevance of a recently developed theory of non-linear representations of Lie groups to the problem of linearization of non-linear field equations covariant under the action of a Lie group is discussed. Some basic definitions and results of this theory are then summarized. The article ends with a discussion of the physical meaning, consequences, and further possible developments of the non-linear representation theory.     

THE NON-LINEAR DYNAMIC BEHAVIOR OF AN ELASTIC LINKAGE MECHANISM WITH CLEARANCES

In terms of Newton two-state model, by choosing two sets of generalized co-ordinates, this paper develops a unified dynamic model between the separation and collision process for the elastic linkage mechanism. This model incorporates the effects of rigidity and elasticity coupling and the angular velocity of crank is assumed to be variable in the operation. In addition, this paper provides a more simple and practical numerical solution method for convenient analysis. Through an example, the dynamic responses of the elastic linkage mechanism with clearances are analyzed, both the effects of elasticity and clearance on the dynamic behaviors of the mechanism are analyzed simultaneously and the non-linear behaviors caused by the clearance joints are analyzed by the dynamic model of rigid mechanism.     

A new orthogonality relationship for orthotropic thin plate theory and its variational principle

While Hamiltonian system was led to solution of elastic theory a symplectic system-atic methodology for theory of elasticity was established and a symplectic orthogonality relationship was presented[1,2]. For two-dimensional theory of elasticity a new dual vec-tor and a new dual differential matrix were presented by putting the old dual vector[1] in a new order. It was discovered for isotropic materials that the symplectic orthogonality relationship may be decomposed into two independent and s…     

Dynamic non-linear response of beams subjected to impact loads

A non-linear theory is presented for plane deformation of beams which allows for longitudinal stretching as well as for cross-sectional stretching and shearing. The exact strain measures for this theory are also deduced. The longitudinal and flexural motions are coupled in the theory. If the cross section is constrained from stretching, the resulting theory may be classified as a non-linear Timoshenko beam theory. The equations of the latter theory are used to study the motion of beams under impact loads.     

Structure and energy of the 90 degrees partial dislocation in diamond: a combined Ab initio and elasticity theory analysis

The core structure and stability of the 90 degrees partial dislocation in diamond is studied within isotropic elasticity theory and ab initio total energy calculations. The double-period reconstruction is found to be more stable than the single-period reconstruction for a broad range of stress states. The analysis of the ab initio results shows further that elasticity theory is valid for dislocation spacings as small as 10-20 A, thus allowing ab initio calculations to provide reliable parameters for continuum theory analysis.     

On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions

We investigate linear theories of incompatible micromorphic elasticity, incompatible microstretch elasticity, incompatible micropolar elasticity and the incompatible dilatation theory of elasticity (elasticity with voids). The incompatibility conditions and Bianchi identities are derived and discussed. The Eshelby stress tensor (static energy momentum) is calculated for such inhomogeneous media with microstructure. Its divergence gives the driving forces for dislocations, disclinations, point defects and inhomogeneities which are called configurational forces.     

On correct nonlocal generalized theories of elasticity

The paper discusses nonlocal elasticity theories among which are models of media with defect fields, gradient elasticity theories, and hybrid nonlocal elasticity theories. Gradient theories are analyzed, and their correctness properties are examined. Applied theories that satisfy the correctness conditions are developed, and known applied gradient theories are verified for the correctness properties. A new nonlocal generalized theory has been developed for which the operator of balance equations is represented as the product of the equilibrium operator of classical elasticity theory and the Helmholtz operator. It is shown that this theory is one-parameter and is the only representative of hybrid models constructed by a complete system of equations for forces and moments. Unlike classical elasticity that is free from scale parameters characterizing the internal material structure, nonlocal elasticity theories naturally incorporate these parameters. That is why they are suitable for the modeling of scale effects and find application in the solution of numerous applied problems for heterogeneous structures with developed phase interfaces where the degree of influence of scale effects depends on the density of phase boundaries. Nonlocal continuum models are especially attractive for modeling the properties of various micro/nanostructures, elastic properties of composites and structured materials with submicron- and nanosized internal structures in which effective properties are to a great extent defined by the scale effects (short-range interaction effects of cohesion and adhesion). Generalized elasticity theories even for isotropic materials contain many additional physical constants that are difficult or impossible to determine experimentally. Applied models with a small number of additional physical parameters are therefore of great interest. However, the reduction of nonlocal theories aimed at reducing the number of additional parameters is a nontrivial task and may lead to incorrect theories. The goal of this paper is to study the symmetry properties in gradient theories, to analyze the correctness of gradient theories, and to develop applied one-parameter elasticity theories.