Geometry of Material Space: Its Consequences in Modern Computational Means

Applications of the concepts of material manifold, pseudo-momentum and Eshelby stress (canonical « material » momentum and stress) to efficient numerical schemes in the thermomechanics of solids are given. These schemes are that of the finite-element method whose uncritical application may cause the appearance of spurious configurational forces, that for the finite-element method where the balance of canonical momentum provides a powerful tool to study the accuracy of the constructed scheme, natural boundary conditions in gradient theories, and a perturbational approach to localized nonlinear waves, and that of the finite-volume method which seems to be particularly well adapted to treat the numerics of wave-like motions in thermomechanical theories of materials. The latter method here is akin to a continuous cellular automaton

Thermodynamics of Discrete Systems and Martensitic Phase Transition Simulation

A thermomechanical approach for the modelling of the phase-transition front propagation in solids is described for the class of thermoelastic phases. This description is based on the balance laws of  continuum mechanics in the reference configuration and the thermodynamics of discrete systems. Contact quantities are introduced following the basic concepts of the thermodynamics of discrete systems. The values of the contact quantities are determined within a finite-volume numerical scheme based on a modification of the known wave-propagation algorithm. No explicit expression is used for the kinetic relation governing the phase transition front propagation. All the needed information is extracted from the thermodynamic consistency conditions for adjacent discrete elements. It is shown that the developed model captures the experimentally observed velocity difference which appears because of impact-induced phase transformation

Unconstrained Hamiltonian formulation of General Relativity with thermo-elastic sources

A new formulation of the Hamiltonian dynamics of the gravitational field interacting with(non-dissipative) thermo-elastic matter is discussed. It is based on a gauge condition which allows us to encode the six degrees of freedom of the ``gravity + matter''-system (two gravitational and four thermo-mechanical ones), together with their conjugate momenta, in the Riemannian metric q_{ij} and its conjugate ADM momentum P^{ij}. These variables are not subject to constraints. We prove that the Hamiltonian of this system is equal to the total matter entropy. It generates uniquely the dynamics once expressed as a function of the canonical variables. Any function U obtained in this way must fulfil a system of three, first order, partial differential equations of the Hamilton-Jacobi type in the variables (q_{ij},P^{ij}). These equations are universal and do not depend upon the properties of the material: its equation of state enters only as a boundary condition. The well posedness of this problem is proved. Finally, we prove that for vanishing matter density, the value of U goes to infinity almost everywhere and remains bounded only on the vacuum constraints. Therefore the constrained, vacuum Hamiltonian (zero on constraints and infinity elsewhere) can be obtained as the limit of a ``deep potential well'' corresponding to non-vanishing matter. This unconstrained description of Hamiltonian General Relativity can be useful in numerical calculations as well as in the canonical approach to Quantum Gravity.Comment: 29 pages, TeX forma

Generating elastic solutions of the Einstein field equations from the Schwarzschild vacuum solution

PreprintThe problem of generating solutions of the Einstein field equations with an elastic energy-momentum tensor from the Schwarzschild vacuum solution by means of conformal transformations is analyzed. Applying the formulation of relativistic elasticity, suitable conformal factors are obtained for static and non-static elastic spacetime configurations and particular solutions are presented. This work shows that the technique used here permits generating new elastic matter solutions from a vacuum spacetime.The author thanks support from FCT ("Fundacao para a Ciencia e Tecnologia"), through the projects UID/MAT/00013/2013 and PTDC/MAT-ANA/1275/2014

Nonequilibrium Thermodynamics of Amorphous Materials III: Shear-Transformation-Zone Plasticity

We use the internal-variable, effective-temperature thermodynamics developed in two preceding papers to reformulate the shear-transformation-zone (STZ) theory of amorphous plasticity. As required by the preceding analysis, we make explicit approximations for the energy and entropy of the STZ internal degrees of freedom. We then show that the second law of thermodynamics constrains the STZ transition rates to have an Eyring form as a function of the effective temperature. Finally, we derive an equation of motion for the effective temperature for the case of STZ dynamics.Comment: 8 pages. Third of a three-part serie

The Slow March towards an Analytical Mechanics of Dissipative Materials

The concern of this work is the derivation of material balance laws for the Green-Naghdi (G-N) theory of dissipationless thermoelasticity. The lack of dissipation allows for a physically meaningfal variational formulation which is used for the application of Noether’s theorem. The balance laws on the material manifold are derived and the extact conditions under which they hold are rigorously studied

An approach to anomalous diffusion in the n-dimensional space generated by a self-similar Laplacian

We analyze a quasi-continuous linear chain with self-similar distribution of harmonic interparticle springs as recently introduced for one dimension (Michelitsch et al., Phys. Rev. E 80, 011135 (2009)). We define a continuum limit for one dimension and generalize it to $n=1,2,3,..$ dimensions of the physical space. Application of Hamilton's (variational) principle defines then a self-similar and as consequence non-local Laplacian operator for the $n$-dimensional space where we proof its ellipticity and its accordance (up to a strictly positive prefactor) with the fractional Laplacian $-(-\Delta)^\frac{\alpha}{2}$. By employing this Laplacian we establish a Fokker Planck diffusion equation: We show that this Laplacian generates spatially isotropic L\'evi stable distributions which correspond to L\'evi flights in $n$-dimensions. In the limit of large scaled times $\sim t/r^{\alpha} >>1$ the obtained distributions exhibit an algebraic decay $\sim t^{-\frac{n}{\alpha}} \rightarrow 0$ independent from the initial distribution and spacepoint. This universal scaling depends only on the ratio $n/\alpha$ of the dimension $n$ of the physical space and the L\'evi parameter $\alpha$.Comment: Submitted manuscrip

Numerical Simulation of Waves and Fronts in Inhomogeneous Solids

Dynamic response of inhomogeneous materials exhibits new effects, which often do not exist in homogeneous media. It is quite natural that most of studies of wave and front propagation in inhomogeneous materials are associated with numerical simulations. To develop a numerical algorithm and to perform the numerical simulations of moving fronts we need to formulate a kinetic law of progress relating the driving force and the velocity of the discontinuity. The velocity of discontinuity is determined by means of the non-equilibrium jump relations at the front. The obtained numerical method generalizes the wave-propagation algorithm to the case of moving discontinuities in thermoelastic solids

Nonequilibrium Thermodynamics of Amorphous Materials II: Effective-Temperature Theory

We develop a theory of the effective disorder temperature in glass-forming materials driven away from thermodynamic equilibrium by external forces. Our basic premise is that the slow configurational degrees of freedom of such materials are weakly coupled to the fast kinetic/vibrational degrees of freedom, and therefore that these two subsystems can be described by different temperatures during deformation. We use results from the preceding paper on the nonequilibrium thermodynamics of systems with internal degrees of freedom to derive an equation of motion for the effective temperature and to learn how this temperature couples to the dynamics of the system as a whole.Comment: 7 pages. Second of a three-part serie