Search

In this part we present five different situations in Mechanics, holonomic and non-holonomic, in Continuum Thermodynamics and in material Science where non-commuting variations appears as a necessary tool for the formulation of Variational Equations. In all five situations considered in this chapter tensor K is related to (or even defined by) some geometrical structure present in the configurational bundle-affine connection, non-holonomic frames, absolute parallelism and its torsion.

In the year 2001, Spanish mathematicians C.Murial and J.Romero, introduced the new procedure of prolongations of vector fields from the space Y, of a configuration bundle π : Y → R to the jet spaces J k (π) (λ-prolongations), new class of “symmetries” (λ-symmetries) for the ordinary differential equations and the new method of the symmetry reduction of the ordinary differential equations ([102]) with the λ-symmetries corresponding to the λ-prolongations.

Differential equations are functional relations between the dynamical variables (dependant variables) and their derivatives. These relations may also include independent variables.

Noether Theory of Symmetries and Conservation laws is a beautiful chapter in mathematical physics. Here we present the form this theory take place in the presence of non-commuting variations. Shortly speaking, in the presence of non-commutative variations, Noether laws becomes balance (not conservation) laws.

In this Chapter we present two applications of the method of non-commuting variables in thermodynamics and material science. As the first application, we present the scheme allowing to obtain Entropy Balance as the Euler-Lagrange equation using the scalar field of thermasy - a quantity introduced by H.Helmholtz (see [58]) and used later in relativistic thermodynamics (D.van Dantzig, see vL,VD), thermoelasticity (A.Green and Nachdi) and Continuum Thermodynamics (G.Maugin).

In this section we introduce the Differentiable manifolds, their mappings and present some examples of these objects.

In this work we apply infinitesimal variational calculus to the systems of balance equations. We determine a class of the exterior n+(n+1)-forms Θ on the jet bundle of infinite order over a configurational bundle π:Yn+m→Xn similar to the class of Lepage n-forms. Systems of differential equations obtained in the way similar to one used in the Lagrangian field theory, include the Euler–Lagrange equations corresponding to a Lagrangian functions as well as arbitrary regular systems of balance equations. For a balance system with a symmetry group G we present the Noether balance laws corresponding to the generators of the Lie algebra of the group G.

In this work we present the theory of balance equations of Continuum Thermodynamics (balance systems) in a geometrical form using the Poincare-Cartan formalism of Multisymplectic Field Theory. A constitutive relation C of a balance system BC is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle π : Y → X and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms of the bundle π on the constitutive mappings C is studied and it is shown that the symmetry group Sym(C) of the constitutive relation C acts on the space of solutions of balance system BC. A version of the Noether Theorem for an action of a symmetry group of vertical automorphisms is presented with the usage of conventional multimomentum mapping. The "entropy principle" of Irreducible Thermodynamics (the requirement that the entropy balance should be the consequence of the balance system) is studied for general balance systems. The structure of corresponding (secondary) balance laws of a balance system BC is studied for the cases of different (partial) 1-jet bundles as domains of constitutive laws. Corresponding results may be considered as a generalization of the transition to the dual, Lagrange-Liu picture of Rational Extended Thermodynamics (RET).

The 4-dim intrinsic (material) Riemannian metric G of the material 4-D space-time continuum P is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness., the aging process is modeled as the evolution of the metric G (most importantly of its time component G00) of the material space-time P embedded into 4-D Newtonian space-time with a Euclidean metric. The evolutional equation for the metric G is derived by the classical variational approach. Construction of the Lagrangian for an aging elastic medium and the derivation of a system of coupled elastostatic and aging equations constitute the central part of the work. The external and internal balance laws associated with symmetries of material and physical space-time geometries are briefly reviewed from a new viewpoint.

A model of the self-driven evolution law of a defective anelastic continuum is presented. Two-dimensional examples are discussed and the role of the Clausius—Duhem inequality in imposing constitutive restrictions is investigated.

We introduce some geometrical structures in the phase space of a dynamical system, naturally defined by the system. We study some properties of these structures and discuss their use for the constructing new characteristics of chaotic dynamical behaviour.

Necking is a significant part of the yielding process in many thermoplastics. It starts as strain localization associated with microshear banding and/or cavitations and appears as a domain of oriented (drawn) material, i.e., a “neck”, separated from the domain of original (isotropic) material by a narrow transition zone, which appears as a distinct boundary of the neck region. On further increase of displacement, the neck propagates through the test specimen under constant draw stress. Strain localization such as crazing and shear bending is associated with necking on micro- and sub-microscales. As a result material toughness, i.e., resistance to cracking, as well as durability, i.e., service lifetime under various service conditions, are related to the material ability to necking and specific characteristics of necking process. Necking is manifested in significant changes in a characteristic length scale, e.g., the distance between equally spaced marks in the reference state may increases by factor of 2 in amorphous polymers and up to a factor of 10 in some semicrystalline thermoplastics. There is also a characteristic relaxation time change during the necking. Thus from continuum mechanics viewpoint, the changes of intrinsic material space-time metric are the most fundamental manifestation of necking. Therefore we model necking phenomena as space-time scales transformation and introduce a four-dimensional (4D) Riemannian metric tensor of a material space-time imbedded into 4D Newtonian (laboratory) space-time with a Euclidean metric. Kinetic equation of necking, i.e., evolution equation for material metric tensor is derived using extremal action principle. An example of traveling wave solution for neck propagation in a tensile bar is presented. Analysis of the solution and comparison with experimental observations are discussed.

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space [Formula: see text] of the model over the space X of basic variables. Extended constitutive surfaces ΣS,κ are described as the Legendre submanifolds ΣS of the space [Formula: see text]shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξK are tangent to the surfaces ΣS,κ, introduce conformally Hamiltonian systems μξK where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

The notion of the stress space, introduced by Schaefer [14], and further developed by Kroner [7] in the context of materials free of defects, is revisited. The comparison between the Geometric Theory of Material Inhomogeneities and the Stress Space approach is discussed. It is shown how to extend Kroner’s approach to the case of the material body with inhomogeneities (defects).

In this work we analyze and compare the model of the material (elastic) element and the entropy form developed by Coleman and Owen with that one obtained by localizing the balance equations of the continuum thermodynamics. This comparison allows one to determine the relation between the entropy function S of Coleman–Owen and that one imported from the continuum thermodynamics. We introduce the Extended Thermodynamical Phase Space (ETPS) [Formula: see text] and realize the energy and entropy balance expressions as 1-forms in this space. This allows us to realizes I and II laws of thermodynamics as conditions on these forms. We study the integrability (closure) conditions of the entropy form for the model of thermoelastic element and for the deformable ferroelectric crystal element. In both cases closure conditions are used to rewrite the dynamical system of the model in term of the entropy form potential and to determine the constitutive relations among the dynamical variables of the model. In a related study (to be published) these results will be used for the formulation of the dynamical model of a material element in the contact thermodynamical phase space of Caratheodory and Hermann similar to that of homogeneous thermodynamics.

In this work, we apply the Poincaré–Cartan formalism of Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles [Formula: see text] of the configurational bundle π : Y → X and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system [Formula: see text] is realized as the mapping between a (partial) k-jet bundle [Formula: see text] and the extended dual bundle [Formula: see text] similar to the Legendre mapping of the Lagrangian Field Theory. The invariant (variational) form of the balance system [Formula: see text] corresponding to a constitutive relation [Formula: see text] is studied. Special cases of balance systems — Lagrangian systems of order 1 with arbitrary sources and RET (Rational Extended Thermodynamics) systems are characterized in geometrical terms. The action of automorphisms of the bundle π on the constitutive mappings [Formula: see text] is studied and it is shown that the symmetry group [Formula: see text] of [Formula: see text] acts on the sheaf of solutions [Formula: see text] of balance system [Formula: see text]. A suitable version of Noether theorem for an action of a symmetry group is presented together with the special forms for semi-Lagrangian and RET balance systems. Examples of energy momentum and gauge symmetries balance laws are provided. At the end, we introduce the secondary balance laws for a balance system and classify these laws for the Cattaneo heat propagation system.

In this work we analyze the relation between the multiplicative decomposition $\mathbf F=\mathbf F^{e}\mathbf F^{p}$ of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations - total $\phi$ and the inelastic $\phi_{1}$. We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations $(\phi,\phi_{1})$ and the material metric $\mathbf g$. Finally the dissipative inequality for the materials of this type is presented.It is shown that the conditions of positivity of the internal dissipation terms related to the processes of plastic and metric evolution provide the anisotropic yield criteria.

We study a behavior of the conformal Laplacian operator $\L_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $\L_g$ on such manifolds. We describe the asymptotic of a general solution of the equation $\L_g u = Q u^{\alpha}$ with $1\leq \alpha\leq \frac{n+2}{n-2}$ near each singular point. In particular, we derive the asymptotic of the Yamabe metric near such singularity., Comment: Revision 61 pages

We analyze some aspects of the kinematic theory of non-uniformly defective elastic crystals. Concentrating on the problem of identifying continuous defective lattices possessing the given defectiveness, as defined by the dislocation density tensor, we investigate the relation between the dislocation density tensor and the Lie algebra of vector fields associated with a defective lattice.

A material ageing parameter, i.e. an additional internal variable, is introduced as the conjugate to the Canonical Energy Momentum Tensor. Ageing is manifested in variation of basic material characteristics such as density, moduli of elasticity, yeald stress, strength and toughness.

We analyze mathematical underpinnings of Davini's theory of defective crystals (1) when the defectiveness of a kinematic state may be material point dependent. We show how the underlying space can be identified with a suitably chosen homogeneous space and how the uniformly defective structure is just a special case. 2000 Mathematics Subject Classification: 53Z05, 74B99