On conservation laws in geometrically nonlinear elasto-dynamic field of non-homogenous materials

By applying Noether’s theorem to the Lagrangian density of non-homogenous elastic materials in the so-called Lagrangian framework, conservation laws in geometrically nonlinear elasto-dynamic field have been studied, and a clear picture of relations between the conservation laws in material space and the material balance laws is given. It is found that the mass density and Lamé’s moduli have to satisfy a set of first-order linear partial differential equations, which contain all the symmetry-transformations of space–time based on Newtonian viewpoint of mechanics. The existence and existent forms of conservation laws in material space are governed by these equations. Especially, translation and rotation of coordinates are symmetry-transformations of the Lagrangian density for obtaining both the conservation laws of homogenous material and the material balance laws of non-homogenous material, but change of coordinate scale is not. However, if the mass density and Lamé’s moduli satisfy special equations simplified from those partial differential equations, change of coordinate scale becomes a symmetry-transformation of the Lagrangian density from which a conservation law follows, whereas the associated material balance law does not exist still. An insight into the usability of those equations for constructing conservation laws is presented, and all the non-trivial conservation laws of the functionally graded material (FGM) layer bonded to a substrate are given for mechanical analysis.     

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.     

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.     

Differential equations determining the function that describes precatastrophic behavior of a system

Based on an analysis of the experimental data obtained for various catastrophic phenomena, Malinetskii et al. [1] considered the main parameters of a process preceding the catastrophe and proposed the following function of time that describes this process: I(t) = A + B(t _c ? t)^α[1 + Ccos(θ log(t _c ? t) ? ?)]. The passage to complex quantities and substitution of variables reveal a power-law character of this approximation. Using this approach, differential equations determining the function that describes the precatastrophic behavior are obtained.     

Total energy framework, finite elements, and discretization via Hamilton’s law of varying action

Within the total energy framework which we introduce here for the first time (in contrast to Lagrangian or Hamiltonian mechanics framework), we provide an alternative and have developed in this paper a general numerical discretization for continuum-elastodynamics directly stemming from Hamilton’s law of varying action (HLVA) involving a measurable built-in scalar function, namely, Total Energy ${[{\mathcal{E}}\left({{\boldsymbol{q}},\dot{{\boldsymbol{q}}}}\right): TQ\rightarrow {\mathbb{R}}]}$ . The Total Energy we use herein for enabling the space discretization is defined as the kinetic energy plus the potential energy for N-body systems, and the kinetic energy plus the total potential energy for continuum-body systems. It thereby provides a direct measure and sound physical interpretation naturally, while enabling this framework to permit general numerical discretizations such as with finite elements. In the variational formulation proposed here, we place particular emphasis upon the notion that the scalar function which represents the autonomous total energy of the continuum/N-body dynamical systems can be a crucial mathematical function and physical quantity which is a constant of motion in conservative systems. In addition, we prove that the autonomous total energy possesses the three invariant properties and can be viewed as the so-called total energy version of Noether’s theorem; therefore, the autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts directly emanating from HLVA inherently involving the scalar function, namely, total energy: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as Hamilton’s principle (HP) is routinely used to derive such equations, but without obvious inconsistency via such a principle as explained in the paper; (ii) explain naturally the Bubnov–Galerkin weighted-residual form that is customarily employed for discretization for both space and time, and alternately, (iii) circumvent relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s second law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (i) and (ii) above if one chooses this option, and instead provides new avenues of discretization for continuum-dynamical systems. The present developments naturally embody the weak form in space and time that can be described by a discrete Total Energy Differential Operator (TEDO). Thereby, a novel yet simple, space-discrete Total Energy formulation proposed here only needs to employ the discrete TEDO which provides new avenues and directly yields the semi-discrete ordinary differential equations in time which can be readily shown to preserve the same physical attributes as the continuous systems for continuum-dynamical applications unlike traditional practices. The modeling of complicated structural dynamical systems such as Euler-Bernoulli beams and Reissner–Mindlin plates is particularly shown here for illustration.     

Invariance and first integrals of continuous and discrete Hamiltonian equations

The relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations is considered. The observation that canonical Hamiltonian equations can be obtained by a variational principle from an action functional makes it possible to consider invariance properties of a functional in the same way as done in the Lagrangian formalism. The well-known Noether identity is rewritten in terms of the Hamiltonian function and symmetry operators. This approach, which is based on symmetries of the Hamiltonian action, provides a simple and clear way to construct first integrals of Hamiltonian equations without integration. A discrete analog of this identity is developed. It leads to a relation between symmetries and first integrals for discrete Hamiltonian equations that can be used to conserve structural properties of Hamiltonian equations in numerical implementation. The results are illustrated by a number of examples for both continuous and discrete Hamiltonian equations.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

Energy integrals for the systems with nonholonomic constraints of arbitrary form and origin

This work analyzes energy relations between nonholonomic systems, whose motion is restricted by nonholonomic constraints of arbitrary form and origin. Such constraints can be natural, originating from spontaneous formulation of the problem, or artificial, expressing some program motion in control theory. On the basis of corresponding Lagrange’s equations, a general law of the change in energy d?/dt was formulated for such systems by the help of which it has been shown that here there exist two types of laws of conservation of energy, depending on the structure of work of these reaction forces. Also, the condition for existence of this second type of the law of conservation of energy has been formulated in the form of the system of differential equations. The results obtained are illustrated by a number of examples, with natural nonlinear constraints, as well as with artificial ones that express some program motion.     

Analytical and numerical treatment of a dynamic crack model

We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u  =  u(y, t) and the one-dimensional crack tip trajectory h  =  h(t). We assume that the crack grows straight. Based on a paper of Nicaise-Sändig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t  =  t _i . The dynamic stress intensity factor will be extracted at t  =  t _i . Its knowledge allows us to compute the crack-tip motion h(t _i+1) with corresponding nonuniform crack speed assuming (t _i+1 ? t _i ) is small. Now, we start from the cracked configuration at time t  =  t _i+1 and repeat the steps. The wave displacements are computed with the FEM-package PDE2D. Some numerical examples demonstrate the proposed method. The influence of finite length of the crack and finite size of the sample on the dynamic stress intensity factor will be discussed in detail.     

Finite element formulations using the theorem of power expended: total energy framework with a differential formulation

Traditional practices involving variational calculus have historically dominated most finite element formulations to-date, and have no doubt served as indispensable tools. Besides these practices, our recent contributions in Acta Mechanica (Har and Tamma, 2009, in press) described new alternatives and developments emanating from Hamilton’s Law of Varying Action (HLVA) as a starting point with a measurable built-in scalar function, namely, the Total Energy. The associated framework (in contrast to Lagrangian or Hamiltonian mechanics framework) demonstrated certain new advances, and also provided some fundamental insight into explaining traditional practices of finite element discretization. Here we additionally provide other advances, new directions, and viable alternatives in contrast to all these past practices which routinely employ variational concepts. In particular, focusing on elastodynamics applications, in this paper we provide for the first time finite element formulations stemming instead from a differential formulation and the theorem of power expended with a measurable built-in scalar function, namely, the Total Energy ${[\mathcal{E}({\varvec q},\dot{{\varvec q}}):TQ\to \mathbb{R}]}$ , as a starting point to capitalize on certain added advantages. The autonomous total energy has time/translational/rotational symmetries for the continuum/N-body dynamical systems. The proposed concepts: (i) can be shown to yield the same governing mathematical model equations of motion that are continuous in space and time together with the natural boundary conditions just as balance laws such as linear momentum or Hamilton’s principle are routinely used to derive such equations, but without resorting to any variational concepts, or approaches such as variational principles, (ii) explain naturally how the classical Bubnov–Galerkin weighted-residual form that is customarily employed for discretization can be readily constructed for both space and time, and alternately, (iii) circumvents relying on traditional practices of conducting numerical discretizations starting either from the balance of linear momentum (Newton’s law) involving Cauchy’s equations of motion (governing equations) arising from continuum mechanics or via (1) and (2) above, and instead provides new avenues of discretization for continuum-dynamical systems. For illustration, numerical discretizations are presented for the modeling of complicated structural dynamical systems.     

Kinetic energy of incompressible microstretch fluid in a domain bounded by rigid walls

The basic physical quantities of microstretch flow are the velocity vector (q?), the microrotation vector (v?) and the microstretch (v), the last quantity being a scalar field signifying the stretch or contraction experienced by the local fluid element. The kinetic energy T of the flow over a domain has contributions T₁, T₂, T₃ one from each of the above three quantities q?, v?, and v. It is shown that Sgn (dT/dt) = −1 and that T(t) ? T(t₀) exp [−σ(t−t₀)] for 0 < t₀ ? t. The (positive) number σ depends on the material constants of the flow and also on the geometry of the domain.     

Magnetotransport Properties,Thermally Activated Flux Flow,and Activation Energies in Ba(Fe0.95 Ni0.05)2As2 and Ba(Fe0.94 Ni0.06)2As2 Superconductors

Thermally assisted flux flow (TAFF) is studied in bulk Ba(Fe_0.95 Ni_0.05)₂As₂ (T _c = 20.4 K) and Ba(Fe_0.94 Ni_0.06)₂As₂ (T _c = 18.5 K) superconductors by transport measurements in magnetic fields up to 18 T. In addition, the upper critical field μ ₀ H _c2(0) and the coherence length ξ(0) are determined. The data is analyzed in the context of the widely accepted Anderson-Kim model and Fischer model. The onset TAFF temperature and the crossover temperature T _x from TAFF to flux flow are determined. The flux pinning activation energy U is modeled as U(T,H) = U ₀(H) f(t) where f(t) is some temperature function and the modified Anderson-Kim model is used to extract U ₀, which is graphed as a function of magnetic field μ ₀ H near T _c. The resistive regime is observed, which is caused by fluctuations. Fisher’s model is applied to determine the glass melting transition temperature; it occurs in the upper TAFF state and not in the expected zero-resistivity vortex solid regime. Furthermore, the resistive transition width is proportional to μ ₀ H, in contrast to Tinkham’s prediction. The H-T phase diagram is drawn.     

On the functional forms of activity coefficients and their properties

A complete system of partial differential equations is obtained for the logarithms of activity coefficients of real chemical mixtures. All solutions of these equations are obtained under mild regularity conditions and the conditions that the limit of ln (f_α), as n_α/n tends to unity, is equal to zero. It is shown that all of the functions ln (f_α) can be determined if any one of them is a known function of temperature, pressure and the independent mole fractions. These results are used to compute the deviations of partial and total volume, entropy and specific heats at constant pressure, and the laws of mass action from what would be given for ideal mixtures. Conditions for static and dynamic stability of equilibrium are obtained, and a forward time integration procedure is given for satisfaction of the laws of mass balance and the full form of the laws of mass action.     

Extremum and variational principles for elastic and inelastic media with fractal geometries

This paper further continues the recently begun extension of continuum mechanics and thermodynamics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution lengthscale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. Here we first generalize the principles of virtual work, virtual displacement and virtual stresses, which in turn allow us to extend the minimum energy theorems of elasticity theory. Next, we generalize the extremum principles of elasto-plastic and rigid-plastic bodies. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries.     

An influence function for the intensity factor in tensile fracture

A semi-infinite plane-strain tensile crack occupies the region z = 0, ? ∞ < x < l(t) of an elastic medium. The faces of the crack are loaded symmetrically with an arbitrary time dependent load ($\text{σ}{}_{\mathrm{zz}}\text{= ?(x, t)}$. We derive a closed form expression for a kernel function K(l; t₁; x, t) such that the stress-intensity factor k(L; t₁) at the point (l (t₁), 0, t₁) is given by

$\text{k}\text{(l;t}{}_1\text{)= ∫}\text{K}\text{(l;t}{}_1\text{x, t)?(x, t)}\text{d}\text{x}\text{d}\text{t}$

the domain of integration being a suitable characteristic triangle in the x, t plane. Because of the special form of K we are able to recover Freund's 1973 result that, given f(x, t), k(i;t_l) depends upon/only through l(t_l) and l'(t_l), the latter through a multiplicative factor k(l').     

Limiting oxygen partial pressure of LaMnO3 phase

Oxygen partial pressure at the decomposition of LaMnO₃(s) has been determined by measuring the dependence of the isothermal electrical conductivity on oxygen partial pressure from 1173 to 1473 K. The Gibbs free energy of formation of LaMnO₃(s) from La₂O₃(s) and MnO(s) was obtained as ΔG⁰_f,T = ?130.9 + 0.0329 T kJ/mol.     

1/f noise in a model of intersecting phase transitions

A mathematical model for the appearance of fluctuations with a spectrum inversely proportional to the frequency (flicker or 1/f noise) upon the intersection of phase transitions is proposed. A system with a two-valley potential, whose dynamics are described by two coupled nonlinear Langevin equations that transform Gaussian δ-correlated noise (white noise) into two modes of stochastic fluctuations with spectra having 1/f ^μ and 1/f ^v frequency distributions, where μ ≈ 1 and 1.5 ? v ? 2, is considered.     

Double power law for basic creep of concrete

The dependence of creep on load duration (t?t′) as well as age at loading t′ is described by the law [1+? ₁ t′^-m (t-t′) ⁿ ]/E ₀ in which m, n,? ₁ E₀=material parameters which are determined from test data by optimization techniques. The law is limited to basic creep, but with different values of material parameters it can also describe drying creep up to a certain time. The previous formulations are extended by introducing the age dependence. This also enhances the reliability in long-term extrapolation of creep data. Substituting t?t′=0.001 day, the law also yields the correct age dependance of the conventional elastic modulus, E. If E₀, which is much larger than E, were replaced by E (as implied by previous power laws without age dependence), the age dependence of creep curves obtained by data analysis would be more scattered, the age dependence of E would have to be described by a separate formula, and more material parameters would be necessary to fit test data. The simplicity of the double power law is a major advantage for statistical evaluation of test data.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.