This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

The term fractal was coined by Benoît Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with prefractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integerorder) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton's principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

We address extreme loading of structures using peridynamics. The peridynamic model is a theory of continuum mechanics that is formulated in terms of integro-differential equations without spatial derivatives. It is a nonlocal theory whose equations remain valid regardless of fractures or other discontinuities that may emerge in a body due to loading. We review peridynamic theory and its implementation in the EMU computer code. We consider extreme loadings on reinforced concrete structures by impacts from massive objects. Peridynamic theory has been extended to model composite materials, fluids, and explosives. We discuss recent developments in peridynamic theory, including modeling gases as peridynamic materials and the detonation model in EMU. We then consider explosive loading of concrete structures. This work supports the conclusion that peridynamic theory is a physically reasonable and viable approach to modeling extreme loading of structures.