Stochastic Mechanics and Fractals

Within or Beyond the Entropy Inequality
Erscheinungsjahr 2027
ISBN: 978-3-032-41448-9
Verlag: Springer

Buch, Englisch, 552 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: Interdisciplinary Applied Mathematics

Stochastic mechanics is not simply the addition of randomness to continuum mechanics; rather, it provides a framework linking continuum mechanics, statistical physics, homogenization, random fields, fractals, and multiscale phenomena. develops this perspective for complex media in which randomness, multiscale geometry, and nonclassical continuum effects are intrinsic rather than perturbative.

The book addresses porous materials, fractured solids, composites, turbulent flows, and odd passive and active media, showing how statistical variability, scale dependence, and generalized continuum descriptions can be incorporated consistently into balance laws, constitutive relations, and variational principles. Part I develops the foundations of stochastic mechanics, classical and micropolar continua, and scale-dependent homogenization. Part II treats random processes and scalar and tensor-valued random fields, including fractal scaling, persistence, and long-range dependence. Part III develops stochastic and multiscale descriptions of cracks, damage, and elastic–inelastic transitions. Part IV uses fluctuation theorems to examine finite-scale departures from the classical entropy inequality and their disappearance under appropriate averaging, including molecular and macroscopic granular systems and extensions to odd matter. Part V develops continuum models for fractal and random porous media, including non-integer-dimensional spaces and fractal and fractional operators.

Three appendices provide concise introductions to random-media models, fractals, and random processes for graduate coursework and self-study. The book is intended for researchers and graduate students in continuum mechanics, applied mathematics, statistical physics, materials science, and related areas.

Zielgruppe

Research

Autoren/Hrsg.

Ostoja-Starzewski, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

From deterministic to stochastic mechanics.- Scaling from SVE to RVE.- Scalar random fields (RFs).- Tensor-valued random fields (TRFs).- Stochastic damage phenomena.- Randomness and fractals at elastic-inelastic transitions.- Spontaneous (apparent) violations of the Second Law.- Stochastic thermomechanics with violations of entropy inequality.- Non-integer dimensional space (NIDS) approach.- Applications of NIDS.- Fractal, Fractional, Ambit, and Markov Concepts in Mechanics.

Über Autor(innen)

Martin Ostoja-Starzewski is Research Professor of Mechanical Science and Engineering and Richard W. Kritzer Faculty Scholar at the University of Illinois Urbana-Champaign, and Research Professor of Computer Science and Mathematics at Cracow University of Technology. Previously, he (co-)authored four books: “Microstructural Randomness and Scaling in Mechanics of Materials,” “Thermoelasticity with Finite Wave Speeds,” “Tensor-Valued Random Fields for Continuum Physics,” and “Random Fields of Piezoelectricity and Piezomagnetism: Correlation Structures.” He has been the editor-in-chief of the Journal of Thermal Stresses and served on the editorial boards of numerous journals, including Acta Mechanica, Computer Methods in Applied Mechanics and Engineering, Fractal & Fractional, Journal of Applied Mechanics, Mathematics and Mechanics of Complex Systems, Mechanics Research Communications, and Probabilistic Engineering Mechanics. He was a Timoshenko Distinguished Visitor at Stanford University and a Rothschild Distinguished Visiting Fellow at the Isaac Newton Institute, University of Cambridge.