The book was published in the Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2019. The highly acclaimed series of monographs provides introductory accounts of specialized topics in mathematical and theoretical physics for graduate students and research workers. The monographs in this series are relevant for a broad audience interested in continuum physics, providing physical applications for mathematicians and mathematical tools for physicists. Subject areas covered include cosmology, astrophysics, relativity theory, particle physics, quantum theory, nuclear physics, statistical mechanics, condensed matter physics, plasma physics, and the theory of chaos.
Ostoja-Starzewski’s book addresses a challenge posed by many areas of continuum physics/mechanics: What are the most general, admissible statistically homogeneous and isotropic tensor-valued random fields (TRFs)? Previously, only the TRFs of rank 0 were completely described. This book assembles a complete description of such fields in terms of one- and two-point correlation functions for tensors of ranks 1 through 4. Working from the standpoint of invariance of physical laws with respect to the choice of a coordinate system, spatial domain representations, as well as their wavenumber domain counterparts, are rigorously given in full detail.
The book also discusses an introduction to a range of continuum theories requiring TRFs, an introduction to mathematical theories necessary for the description of homogeneous and isotropic TRFs, and a range of applications. Topics covered in the latter include TRFs in classical and micropolar continuum mechanics, TRFs of constitutive responses and scaling laws, microphysics/mechanics-based stochastic partial differential equations (elliptic and hyperbolic), and TRFs for damage mechanics.