"Material" mechanics of materials

The paper outlines recent developments and prospects in the application of the continuum mechanics expressed intrinsically on the material manifold itself. This includes applications to materially inhomogeneous materials physical effects which, in this vision, manifest themselves as quasi-in homogeneities, and the notion of thermo dynamical driving force of the dissipative progress of singular point sets on the material manifold with special emphasis on fracture, shock waves and phase-transition fronts.

THERMOMECHANICS OF HETEROGENEOUS MATERIALS WITH WEAKLY NONLOCAL MICROSTRUCTURE

A unified thermomechanical framework is presented for the theory of mechanically elastic materials the physical description of which requires the consideration of additional variables of state and of their gradients (weak nonlocality). This includes both the case of additional degrees of freedom carrying their own inertia and the case of diffusiw internal variables of state. In view of practical applications to fracture and propagation of phasetransition fronts. special attention is paid to the construction and immediate consequences of the equations of balance of canonical momentum (on the material manifold) and energy at regular points and at jump discontinuities. In particular. the general expression of the dissipation at, and of the driving force acting on. phase-transition fronts is formally obtained in such a broad framework. Brief applications include thermoelastic conductors (e.g. shape-memory alloys) and elastic ferromagnets in which both spin inertia and ferromagnetic exchange forces (magnetic ordering) are taken into account

Introduction to the thermomechanics of configurational forces

Configurational forces are thermodynamic conjugates to irreversible material body evolutions such as extension of cracks, progress of phase-transition fronts, movement of shock waves, etc. They do correspond to a change of material configuration. Accordingly, their realm is the material manifold of a body. Furthermore, they acquire a physical meaning only in so far as they contribute to the global dissipation. Therefore, the present contribution of a pedagogical nature proposes a primer introduction to the thermodynamics of configurational forces. To that purpose, we first introduce a consistent thermomechanics of general deformable continua on the material manifold (and not in physical space). This is achieved in a canonical manner by full projection of the balance equation of momentum onto the material manifold and constructing in parallel a formally consistent expression of the energy conservation. Then various configurational forces such as those appearing in inhomogeneous bodies, at the tip of a propagating crack, at the surface of a propagating phase-transition front, or of a shock wave, and those due to local structural rearrangements (plasticity, damage, growth), are examined from the point of view of their dissipated power

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

Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

We deduce the dynamic frequency-domain-lattice Green's function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic I D exponentially graded material. The discrete model yields closed form expressions for the N x N Green's function for an arbitrary number N = 2,...,infinity of particles of the chain. Utilizing this Green's function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a I D exponentially graded elastic material. As a special case the well-known expressions for the Green's function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green's function recovers the Green's function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation. (C) 2008 Elsevier Ltd. All rights reserved