Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.     

From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed     

Statistically equilibrium two-dimensional vortices

Equilibrium statistical mechanics is used for describing two-dimensional vortices in an unbounded incompressible ideal fluid. Both the energy and angular momentum integrals and a set of invariants are taken into account. The latter follows from the condition that any vorticity distribution can be obtained from an initial distribution by a differentiable areas-preserving transformation. The equations for the statistically equilibrium vorticity and passive admixture distributions are derived. It is argued that taking subsidiary invariants into account weakens the arbitrariness associated with the choice of a finite-dimensional approximation of the flow. The case in which the vorticity cloud behaves like a thermodynamic system undergoing an ordering phase transition is discussed.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–55, September–October, 1995.     

Strain-energy density function for rubberlike materials

The strain-energy density function surface for the rubber tested by L. R.G. Treloak (1944a) is determined from bis stress-strain data. The data were given for the three pure homogeneous strain paths of simple elongation, pure shear, and equi-biaxial extension of a thin sheet. The surface is formed by plotting calculated points of the strain-energy function above a plane having the first and second strain invariants as rectangular cartesian coordinates. The strain-energy function is expressed as a double power series in the invariants expanded about the zero energy state which is the origin of coordinates. An analysis of this experimentally derived surface provides the information required for the rational selection of terms and the determination of the coefficients in the series expansion, thus defining a function within the Rivlin-type formulation. The function so determined is tested by employing it in the appropriate constitutive formulae to compute stresses for comparison with experimental values. Another test is made by utilizing the function to predict shapes of an inflated membrane for comparison with experimentally observed shapes. Excellent agreement between prediction and experiment is found. A second demonstration is given for another rubber tested by D.F. Jones and L.R.G. Treloar (1975). Again, excellent results are obtained.     

Laplace Type Invariants for Parabolic Equations

The Laplace invariants pertain to linear hyperbolic differentialequations with two independent variables. They were discovered byLaplace in 1773 and used in his integration theory of hyperbolicequations. Cotton extended the Laplace invariants to ellipticequations in 1900. Cotton's invariants can be obtained from the Laplaceinvariants merely by the complex change of variables relating theelliptic and hyperbolic equations.To the best of my knowledge, the invariants for parabolic equations werenot found thus far. The purpose of this paper is to fill this gap byconsidering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found forparabolic equations.     

A method for finding the invariants and exact solutions of coupled non-linear differential equations with applications to dynamical systems

The polynomial invariants of (a set) non-linear differential equations are found by using a direct approach. The integrability of these invariants deserves the integrability of the given set of coupled differential equations. As applications, the Lorenz and Rikitake sets, among others, are studied. New invariants are obtained.     

A discussion about scale invariants for tensor functions

It is found that in some cases the complete and irreducible scale invariants given by Ref.[1] are not independent. There are some implicit functional relations among them. The scale invariants for two different cases are calculated. The first case is an arbitrary second order tensor. The second case includes a symmetric tensor, an antisymmetric tensor and a vector. By using the eigentensor notation it is proved that in the first case there are only six independent scale invariants rather than seven as reported in Ref.[1] and in the second case there are only nine independent scale invariants which are less than that obtained in Ref.[1].     

Lie symmetries,symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.     

Constitutive structure in coupled non-linear electro-elasticity: Invariant descriptions and constitutive restrictions

A constitutive framework for electro-sensitive materials in the context of non-linear elasticity is analyzed. Constitutive equations are given in terms of energy functions that depend on several invariants. The study includes both the analysis of the invariants, which are present in the energy functions, and the analysis of constitutive restrictions that have to be obeyed by the constitutive functions. Isotropic as well as non-isotropic electro-sensitive elastomers are studied. The set of invariants that describe each material model is analyzed under two homogeneous deformations: (i) an uniaxial elongation and (ii) a simple shear deformation. These deformations are chosen since they are relevant to specific experiments, from which one may try to fit constitutive equations. The constitutive restrictions developed are based on classical ones used for isotropic non-linear elastic materials, in particular, are based on the Baker–Ericksen inequality and the ellipticity condition.     

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional(2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.     

Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

In this article we investigate several models contained in the literature in the case of near-incompressibility based on invariants in terms of polyconvexity and coerciveness inequality, which are sufficient to guarantee the existence of a solution. These models are due to Rivlin and Saunders, namely the generalized polynomial-type elasticity, and Arruda and Boyce. The extension to near-incompressibility is usually carried out by an additive decomposition of the strain energy into a volume-changing and a volume-preserving part, where the volume-changing part depends on the determinant of the deformation gradient and the volume-preserving part on the invariants of the unimodular right Cauchy–Green tensor. It will be shown that the Arruda–Boyce model satisfies the polyconvexity condition, whereas the polynomial-type elasticity does not. Therefore, we propose a new class of strain-energy functions depending on invariants. Moreover, we focus our attention on the structure of further isotropic strain-energy functions.     

On the Modeling of Extension-Torsion Experimental Data for Transversely Isotropic Biological Soft Tissues

For the problem of torsion superimposed on extension of incompressible nonlinearly elastic transversely isotropic circular cylinders, a simple asymptotic analysis is carried out on using a small parameter that reflects the moderate twisting of slender cylinders, which corresponds to a typical testing regime for biological soft tissue. The analysis is carried out for a subclass of strain-energy densities that reflect transversely isotropic material response. On using a four-parameter polynomial expression for the strain-energy density in terms of certain classical invariants, this analysis is shown to be in excellent agreement with experimental data obtained by other authors for rabbit papillary muscles. An explicit condition on the strain-energy density is obtained that determines whether the stretched cylinder tends to elongate or shorten on twisting. For the special case of pure torsion where no extension is allowed, this condition determines whether the classical or reverse Poynting effect occurs. For the rabbit papillary muscles, the theoretical results predict and the experimental results confirm that a reverse Poynting-type effect occurs where the stretched rabbit muscle tends to shorten on twisting.     

Plane mechanics and kinematics of compressible ideal granular materials

Summary In a previous paper (1) proposals were made for the equations which govern the mechanical behaviour in plane strain of an ideal incompressible granular material. In this paper tentative suggestions are made for the extension of this theory to compressible granular materials. The material is envisaged as a mixture of an ideal gas and solid particles. The state of an element of the mixture is determined by its overall density and the mass and volume concentrations of the constituents. It is proposed that the material satisfies a flow condition which relates the two principal invariants of the stress (in two dimensions) and the density. The condition for the stress equations to be hyperbolic is obtained, and for the hyperbolic case a natural interpretation is obtained for the angle of internal friction.For the kinematic behaviour it is proposed that the deformation consists of superposed shearing deformations on the stress characteristic surfaces, as described in (1), together with a superposed dilatation. The equations describing this behaviour are expressed in characteristic form. They reduce to the equations given in (1) when the density is constant.     

The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

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Constitutive equations of thermoplasticity including the third invariant

Constitutive equations describing the nonisothermal deformation of elements of a body along paths of small curvature are formulated taking into account the stress mode. The equations include two scalar functions, one relating the first invariants of the tensors and the other relating the second invariants of the stress and strain deviators. Both scalar functions are nonlinear, dependent on temperature and stress mode, and determined in tests on tubular specimens. The plastic incompressibility condition is validated in uniaxial-tension tests on tubular and solid specimens. The proposed equations are used to design a loading process that differs from the base ones and proceeds at high temperature. The calculated results are compared with experimental data     

The relation between the Valanis-Landel and classical strain-energy functions

Necessary and sufficient conditions are derived for the strain-energy function of an isotropic elastic solid, regarded as a function of the strain invariants, to be expressible in the Valanis-Landel form, both when the material is compressible and when it is incompressible. In the case when the Valanis-Landel strain-energy function is a polynomial in squares of the principal extension ratios, explicit representations as polynomials in the basic isotropic strain invariants are obtained.     

Dynamics of a physical SBT memristor-based Wien-bridge circuit

The invariants of an attractor have been the most used resource to characterize a nonlinear dynamics. Their estimation is a challenging endeavor in short-time series and/or in presence of noise. In this article, we present two new coarse-grained estimators for the correlation dimension and for the correlation entropy. They can be easily estimated from the calculation of two U-correlation integrals. We have also developed an algorithm that is able to automatically obtain these invariants and the noise level in order to process large data sets. This method has been statistically tested through simulations in low-dimensional systems. The results show that it is robust in presence of noise and short data lengths. In comparison with similar approaches, our algorithm outperforms the estimation of the correlation entropy.     

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.     

Distinguishing Periodic and Chaotic Time Series Obtained from an Experimental Nonlinear Pendulum

The experimental analysis of nonlinear dynamical systems furnishes ascalar sequence of measurements, which may be analyzed using state spacereconstruction and other techniques related to nonlinear analysis. Thenoise contamination is unavoidable in cases of data acquisition and,therefore, it is important to recognize techniques that can be employedfor a correct identification of chaos. The present contributiondiscusses the experimental analysis of a nonlinear pendulum, consideringstate space reconstruction, frequency domain analysis and thedetermination of dynamical invariants, Lyapunov exponents and attractordimension. A procedure to construct Poincaré map of the signal ispresented. The analyses of periodic and chaotic motions are carried outin order to establish a difference between them. Results show that it ispossible to distinguish periodic and chaotic time series obtained froman experimental set up employing proper procedures even though noisesuppression is not contemplated.