Fracture Parameter for Thermoplasticity in the Case of Dilatation Symmetry

An extension of the M integral to thermoplasticity related to the dilatational or scaling symmetry is presented. Based on a discrete Lagrangian describing the thermoinelastic system andon Noether's theorem defining the conservation law, the null Lagrangian theorem is applied in order to introduce thermoplasticity. By the use of the divergence theorem, the modified M integral is obtained.     

Constitutive relations of micromorphic thermoplasticity

The constitutive theory of micromorphic thermoplasticity has been formulated in Lagrangian form. The generalized Lagrangian strains, strain rates, temperature, temperature rate and temperature gradients are considered as the independent constitutive variables. Internal variables, including generalized Lagrangian plastic strains, are incorporated. The axioms of objectivity and equipresence are strictly followed. The entropy inequality has been fully utilized to guide the derivations of the constitutive relations. No restrictive assumption has been made to the magnitude of any independent constitutive variables. Finite element formulations for micromorphic thermoplasticity are developed based on the balance laws and the constitutive theory.     

On the first integral of the non-linear shallow-membrane equation via Noether's theorem

In this paper, using the invariance identities of Rund [2], involving the Lagrangian and the generators of the infinitesimal Lie group and then using Noether's theorem, the first integral of the shallow-membrane equation has been obtained. Further, through the repeated application of invariance under the transformation obtained its exact solution, for certain particular values of the parameters involved, is obtained by reducing the first integral to quadrature.     

Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space

This paper focuses on studying Noether’s theorem in phase space for fractional variational problems from extended exponentially fractional integral introduced by El-Nabulsi. Both holonomic and nonholonomic systems are studied. First, the fractional variational problem from extended exponentially fractional integral, as well as El-Nabulsi–Hamilton’s canonical equations are established; second, the definitions and criteria of fractional Noether symmetric transformations and fractional Noether quasi-symmetric transformations are presented which are based on the invariance of El-Nabulsi–Hamilton action under the infinitesimal group transformations; finally, the fractional Noether’s theorem is established, which reveals the inner relationship between a fractional Noether symmetry and a fractional conserved quantity.     

Generalized optical ABCD theorem and its application to the diffraction integral calculation

We generalize the transfer matrix ABCD theorem for paraxial rays of the optical system to skew rays propagated off axis, whether or not the system possesses rotational symmetry. Furthermore, we apply the generalized ABCD theorem to evaluate the diffraction integral matrix elements A-D expressed in terms of the angle eikonal T, with the primary aberrations included. Finally, analysis and numerical calculation are given for propagation of a light beam through the optical system in the case in which spherical aberration and coma are present.     

Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times

A general model for the linear micropolar electro-magnetic thermoelastic continuum based on the hyperbolic heat equation, which is physically more relevant than the classical thermoelasticity theory in analyzing problems involving very short intervals of time and/or very high heat fluxes, is introduced. An integral identity that involves two admissible processes at different instants is established. Uniqueness theorem is proved, with no definiteness assumption on the elastic constitutive coefficients and no restrictions on the electro-elastic coupling moduli, magneto-elastic coupling moduli, and thermal coupling coefficients other than symmetry conditions. The reciprocity theorem is derived, without the use of Laplace transforms. The integral representation formula is obtained in case instantaneous concentrated, time-continuous or time-harmonic loads are applied. The Maysel’s, Somigliana’s and Green’s formulas are derived. The mixed boundary value problem is considered and a system of five singular Fredholm integral equations is obtained. The results for dynamic classical coupled theory can be easy deduced from the given general model formulated for the temperature-rate dependent thermoelasticity.     

Conformal invariance of Mei symmetry for discrete Lagrangian systems

The conformal invariance of the Mei symmetry and the conserved quantities are investigated for discrete Lagrangian systems under the infinitesimal transformation of the Lie group. The difference Euler–Lagrange equations on regular lattices of the discrete Lagrangian systems are presented via the transformation operators in the space of the discrete variables. The conformal invariance of the Mei symmetry is defined for the discrete Lagrangian systems. The criterion equations and the determining equations are proposed. The conserved quantities of the systems are derived from the structure equation governing the gauge function. Two examples are given to illustrate the application of the results.     

The generalized curie principle,the hermann theorem,and the symmetry of macroscopic tensor properties of composites

The symmetry of tensor properties of composite materials is discussed in terms of the generalized Curie principle (including the effect of symmetrizing factors), as well as the Hermann theorem concerning the relationship between the rank of a tensor property and the presence or absence of isotropy in a plane normal to a given symmetry axis. A simple configuration for achieving this transverse isotropy for the elastic behaviour of laminated structures is proposed as an illustration of the practical application of the Hermann theorem. A recently published theorem about the point symmetry of composite objects is shown to be invalid.     

分数阶脉冲微分方程初值问题解的存在性

利用Green函数可以将分数阶微分方程初值问题转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程初值问题解的存在性.本文讨论菲线性分数阶脉冲微分方程初值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Carathéodory条件,利用非紧性测度的性质和M(o)nch,8不动点定理证明解的存在性.     

Force on a defect in non-linear elastic medium

A derivation is given for the force per unit surface of a defect (e.g. a dislocation) in a non-linear elastic medium. The derivation is based on a Lagrangian of the same form as in linear elasticity, introduced in [1]. Principle of energy conservation is deduced on the basis of a particular case of Noether's theorem. Some peculiar properties of the Lagrangian are discussed. In the particular case of a constant Burgers vector the force reduces to the Peach-Koehler force per unit dislocation line, known in the linear theory of elasticity.     

Application of a boundary integral equation to prediction of freezing fronts in soil

A boundary integral equation formulation based on the complex Cauchy integral theorem is applied to two-dimensional soil-water phase change problems encountered in algid soils. The model assumes that potential theory applies in the estimation of heat flux along a freezing front of differential thickness and that quasi-steady-state temperatures occur along the problem domain boundary. Application of the boundary integral formulation to two-dimensional problems results in predicted locations of the freezing front which are highly accurate. Although the proposed formulation is based on the Cauchy integral theorem, similar models may be developed based on other forms of integration equation methods.     

Analysis of a class of nonconservative systems reducible to pseudoconservative ones and their energy relations

This paper analyzes a class of nonconservative systems, whose Lagrangian equations can be reduced to Euler–Lagrangian equations by introducing a new Lagrangian, which is equal to a product of some function of time f(t) and the primary Lagrangian. These equations formally have the same form as for the systems with potential forces, while the influence of nonconservative forces is contained in the factor f(t), and such systems are called pseudoconservative. It is further shown that the requirement for a nonconservative system to be considered as a pseudoconservative is the existence of at least one particular solution of a system of differential equations with unknown function f(t), or their linear combination with suitably chosen multipliers. Further on, the energy relations and corresponding conservation laws of those systems are analyzed from two aspects: directly, on the basis of the corresponding Lagrangian equations and via modified Emmy Noether’s theorem. So, it has been shown, even in two different ways, that there are two types of the integrals of motion, in the form of the product of an exponential factor and the sum of the generalized energy (energy function) and an additional term. For the existence of these integrals of motion, it is necessary and sufficient that there exists at least one particular solution of a partial differential equation, which is in accordance with the Lagrangian equations for the observed problem. The obtained results are equivalent to so-called energy-like conservation laws, obtained via Vujanovi?-Djuki?’s generalized Noether’s theorem for nonconservative systems (Vujanovi? and Jones in: Variational Methods in Nonconservative Phenomena (monograph). Acad. Press, Boston, 1989).     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.     

A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null‐space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so‐called symmetries. The symmetries relevant for rod mechanics are frame‐indifference, isotropy, and uniformity. We show that a completely analogous and self‐contained theory of discrete rods can be formulated in which the arc‐length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc‐length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application. Copyright © 2010 John Wiley & Sons, Ltd.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Thermoplasticity with diffusion in welding problems

A thermomechanical model for the analysis of thermoplasticity with a diffusion term is presented. This model is defined in infinitesimal strain thermoplasticity framework. Thermal hardening is coupled with a diffusion process. The example presents thermodiffusion process in titanium friction welding parts. The migration of hydrogen in the process of thermoplastic deformation is described. Copyright © 2007 John Wiley & Sons, Ltd.     

A complex variable boundary element method: Development

A generalized boundary integral equation method for the solution of the Laplace equation is developed based on the Cauchy integral theorem for analytical complex variable functions. Although the approach is complicated by the utilization of complex variable theory, the resulting model involves direct integration along straight-line boundary segments (elements) rather than using quadrature formulae that are required in current real variable boundary element formulations. Previously published boundary integral equation methods based on the Cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper.     

Uncertainties in Interpolated Spectral Data

Interpolation is often used to improve the accuracy of integrals over spectral data convolved with various response functions or power distributions. Formulae are developed for propagation of uncertainties through the interpolation process, specifically for Lagrangian interpolation increasing a regular data set by factors of 5 and 2, and for cubic-spline interpolation. The interpolated data are correlated; these correlations must be considered when combining the interpolated values, as in integration. Examples are given using a common spectral integral in photometry. Correlation coefficients are developed for Lagrangian interpolation where the input data are uncorrelated. It is demonstrated that in practical cases, uncertainties for the integral formed using interpolated data can be reliably estimated using the original data.     

A variational statement of quasistatic ��rigid-deformable�� contact problems at large strain involving generalized forces and friction

A new statement for a certain contact problem subclass is proposed. The statement is based solely on a weak solution notion for the equilibrium equation and does not utilize both optimization theory (penalty, Lagrangian multipliers and augmented Lagrangian) and mathematical programming. An inverse theorem is proved. Numerical examples for the 2-D case are given. The approach reported in this paper allows to define the law for movement of the rigid body, which acts on the deformable one with the force changeable in compliance with the prescribed program. This is the primary distinction from other works devoted to the contact subjects, and is very important for practical applications where the rigid body is the deforming tool.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.