Sublaminate-based lamination theory and symmetry properties of textile composite laminates

Textile composites have distinct mechanical characteristics different from those of unidirectional composites. Several common weave patterns, 5-harness or 8-harness satin weaves, for example, lack certain symmetry properties common to composites composed of unidirectional layers. The lack of symmetry of woven fabric composites can be drastically reduced, but not totally eliminated, if a laminate consists of one or more flipped pairs, each with one normal layer placed symmetrically about the laminate mid-plane to the flipped layer. To explore more possible combinations in a more general way, a systematic approach is presented by considering a single fabric layer as a sublaminate characterized by a set of stiffness parameters. Mechanical properties of laminates constructed from such sublaminates are obtained by combining the sublaminate stiffness matrices and force and moment resultants. The effects of operations like 90° rotation, flipping, mirroring and their combinations on a sublaminate are discussed. The symmetry properties of weave patterns are classified based on the type of operation necessary for the patterns to be equivalent before and after the operation. The classification of fabric patterns and their coupling coefficients of plate stiffness matrices are found to be dependent on the distribution of crimped regions, or more specifically, the skewness of that distribution. Techniques for achieving symmetry and balance by constructing laminates from flipped and/or rotated pairs are then discussed and found to be, to a certain extent, dependent on the type of equivalence of the weave pattern. It is shown in this study that perfect symmetry cannot be achieved for weave patterns, such as 8-harness satin weaves, with only rotation-flip equivalence. Perfect symmetry and balance can be achieved using as few as four layers, within cell alignment errors, for weave patterns with only mirror-rotation equivalence, like 5-harness satin weaves.     

Relevance of symmetry methods in mechanics of materials

The interest and relevance of symmetry methods as a predictive and systematic methodology in the continuum mechanics of materials is analyzed, relying on a classification of the inherent aspects in terms of the direct, extended direct, and inverse methods. Although being interrelated, these three problems each have a specific argumentation which is separately exposed in the present contribution. The direct problem of finding invariants associated with a given constitutive law for materials, including dissipation, is first envisaged. The abstract formulation of constitutive laws in terms of the state laws and a dissipation potential expressing the evolution of internal state variables is considered, in the framework of irreversible thermodynamics. It is shown that a specific choice of the components of the symmetry vector acting in the space of independent and dependent variables leads to a local invariance condition of the constitutive law fully equivalent to the variational symmetry condition using the rate of the internal energy density. As a specific situation involving this methodology, a time–temperature equivalence principle of polymers is obtained from the requirement of group invariance of the field equations. A validation of this invariance principle is given by a comparison of the modelled master response and the master curve constructed from a set of experimental results at various temperatures. The extended direct method is next presented as a generalization of the direct method, in the sense that a classification of constitutive functions modelling the material behavior is achieved via a symmetry analysis. In the third part of the paper, the inverse problem of constructing a material’s constitutive law exploiting a postulated Lie-group structure is exposed. A constitutive model is then identified which satisfies the symmetries exhibited by the experimental data.     

Quasivelocities and symmetries in non-holonomic systems

This article is concerned with the theory of quasivelocities for non-holonomic systems. The equations of non-holonomic mechanics are derived using the Lagrange–d'Alembert principle written in an arbitrary configuration-dependent frame. The article also shows how quasivelocities may be used in the formulation of non-holonomic systems with symmetry. In particular, the use of quasivelocities in the analysis of symmetry that leads to unusual momentum conservation laws is investigated, as is the applications of these conservation laws and discrete symmetries to the qualitative analysis of non-holonomic dynamics. The relationship between asymptotic dynamics and discrete symmetries of the system is also elucidated.     

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).     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

Multi-cluster dynamics in coupled phase oscillator networks

In this paper, we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function g(?) (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work, we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N = 6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability.     

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.     

Optimal control of a continuum model for single-product re-entrant manufacturing systems

A semiconductor manufacturing system that involves a large number of items and many steps can be modelled through conservation laws for a continuous density variable on a production process. In this paper, the basic hyperbolic partial differential equation (PDE) models for multiple re-entrant manufacturing systems are proposed. However, through numerical examples, the basic continuum models do not perform well for small-scale multiple re-entrant systems, so a new state equation taking into account the re-entrant degree of the product is introduced to improve the basic continuum models. The applicability of the modified continuum model is illustrated through a numerical example. Based on the modified continuous model, this paper studies the optimal control problems for multiple re-entrant manufacturing systems. The gradient of the cost function with respect to the influx is solved by the adjoint approach, and then the optimal influx is computed by the steepest descent method. Finally, numerical examples on optimal influx profiles for steps in demand rate, linear demand rate and periodically varying demand rate are given. The relationships among influx, outflux and demand are also discussed in detail.     

ISovector fields and self-similar solutions for power law creep

Isovector methods are a recently developed technique for systematically investigating properties of the solutions of systems of differential equations. These methods are applied to the nonlinear equations of power law creep with elastic strains under conditions of plane and antiplane strain. Among the results are a family of self-similar solutions which are shown to be the only ones extant for the cases investigated. Some light is also shed upon the existence of conservation laws for these equations, and upon the existence of mappings which transform the governing equations into a set of equivalent linear equations.     

Irreversible thermodynamics of nonlocal systems

Global laws of balance of momentum, moment of momentum and energy, together with local conservation of mass are reduced to point statements involving localization residuals. A fundamental functional inequality is then obtained which reduces to the Clausius-Duhem inequality for local theories. For nonlocal theories, the functional inequality states that the total internal production of heat of a material body at any given time is non-negative. This inequality is reformulated in terms of a functional inequality on a Hilbert space of ordered collections of L₂ functions. The general solution of this functional inequality is obtained and this leads to all admissible constitutive relations. The existence of dissipation potentials and symmetry relations are established for material bodies with admissible constitutive relations. Nonlocal analogues of Maxwell's reciprocity relations are also obtained as well as a proof of consistency with the results of thermostatics. Satisfaction of nonlinear forms of Onsager's reciprocity relations are shown to be equivalent to the requirement that the operators generating the admissible constitutive relations be potential operators. It is also shown the certain functionals of curves in a function space are odd under time reversal if and only if the nonlinear form of Onsager's reciprocity relations are satisfied. Thus, Gurtin's results [10] for processes which may be approximated by linear departures from equilibrium are extended to all processes with admissible constitutive relations. Similar results are established for significantly less restrictive sets of histories than those used by Gurtin and for a wide class of generalizations of the time reversal operator on such histories. Indications are given that satisfaction of invariance under superimposed rigid body motions implies satisfaction of the zero mean conditions for all localization residuals.     

Equilibrium Simulations of 2D Weak Links in p-wave Superfluids

A two-dimensional Ginzburg-Landau theory of weak links in a p-wave superfluid is presented. First we consider the symmetry properties of the energy functionals, and their relation to the conserved supercurrents which play an essential role in the weak link problem. In numerical studies, we use the A and B phases of superfluid ³He. The phases on the two sides of the weak link can be chosen separately, and very general soft degrees of freedom may be imposed as boundary conditions. We study all four inequivalent combinations of A and B which are possible for a hole in a planar wall, including weak links with a pinned A-B interface. In all cases, some illustrative current-phase relations (CPR's) are calculated and the critical currents are mapped. Phase diagrams covering the relevant phase space in zero magnetic field are constructed. The numerical methods are also described in some detail.     

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.     

Symmetry and chaos in the complex Ginzburg-Landau equation - I. Reflectional symmetries

The complex Ginzburg-Landau (CGL) equation on a one-dimensional domain with periodic boundary conditions has a number of different symmetries. Solutions of the CGL equation may or may not be fixed by the action of these symmetries. We investigate the stability of chaotic solutions with some reflectional symmetry to perturbations which break that symmetry. This can be achieved by considering the isotypic decomposition of the space and finding the dominant Lyapunov exponent associated with each isotypic component. Our numerical results indicate that for most parameter values, chaotic solutions that have been restricted to lie in invariant subspaces are unstable to perturbations out of these subspaces, leading us to conclude that for these parameter values arbitrary initial conditions will generically evolve to a solution with the minimum amount of symmetry allowable. We have also found a small region of parameter space in which chaotic solutions that are even are stable with respect to odd perturbations.     

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations. Received: May 26, 2000; revised version: September 6, 2000     

Conservation laws, duality and symmetry loss in solid mechanics

The paper deals with conservation laws which are not of the pure divergence type and thus do not provide a path-independent integral for use in Fracture Mechanics. It is shown that Duality is the right tool to re-establish the symmetry between equations and to provide conservation laws of the pure divergence type. The loss of symmetry of some energetic expressions is exploited to derive a new method for solving some inverse problems. In particular, the earthquake inverse problem is solved analytically. Dedicated to George Herrmann.     

Lagrange identification of absolutely unstable regimes in wakes

Summary The recent theory of a nonlocal micropolar continuum is used to derive explicit expressions for the set of conservations laws. Using the Euclidean group of transformation, the equivalence between conservation laws and Euclidean invariance is demonstrated. It is shown that one of the integrals (J type integral) has a physical meaning of the energy release rate.     

Conservation laws for 3-dimensional compressible Euler equations

The generators of infinitesimal symmetry transformations for the Euler equations, and the corresponding set of adjoint variables are derived. The associated conservation laws are then discussed. A detailed analysis of 1-dimensional flows brings into evidence the connections with current alternative approaches to conservation laws.     

Computation of magnetostatic fields in three-dimensions based on Fredholm integral equations

An integral equation method is described for solving three-dimensional magnetostatic problems involving linear permeability interfaces and current sources. The interface is replaced by magnetic charges in free space to provide an equivalent interface condition. Application has been made to leakage field calculations in transformers. Numerical methods are combinations of analytical formulas and numerical integration; basis functions approximate the charge distribution.     

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.     

General parametric stiffness excitation – anti-resonance frequency and symmetry

Summary Stability investigations on vibration cancelling employing the concept of actuators with general stiffness elements are presented. Systems with an arbitrary number of degrees of freedom with linear spring- and damping elements are considered, that are subject to self-excitation as well as parametric excitation by stiffness variations with arbitrary phase relations. General conditions for full vibration suppression are derived analytically by applying a singular perturbation technique. These conditions naturally lead to the terms of parametric resonance and anti-resonance and enable a stability classification with respect to the parametric excitation matrices and their symmetry properties. The results are compared to former investigations of systems with a single or synchronous stiffness variation in time and geometrical interpretations are given. These basic results obtained can be used for design of a control strategy for actuators with periodically actuated stiffness elements and arbitrary phase relations.