Jun Li

This paper builds on the recently begun extension of continuum thermomechanics to frac-tal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cutoffs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.

This paper studies fractal patterns forming at elastic-plastic transitions in soil-and rock-like materials. Taking either friction or cohesion as nonfractal vector random fields with weak noise-to-signal ratios, it is found that the evolving set of plastic grains (i.e., a shear-band system) is always a monotonically growing fractal under increasing macroscopic load in plane strain. Statistical analysis is used to assess the anisotropy of those shear bands. All the macroscopic responses display smooth transitions, but as the randomness vanishes, they turn into a sharp response of an idealized homogeneous material. Parametric study shows that increasing hardening or friction makes the transition more rapid. In addition, randomness in cohesion has a stronger effect than randomness in friction, whereas dilatation has practically no influence. Adapting the concept of scaling functions, the authors find the elastic-plastic transitions in random Mohr-Coulomb media to be similar to phase transitions in condensed-matter physics: the fully plastic state is a critical point, and with three order parameters (reduced Mohr-Coulomb stress, reduced plastic volume fraction, and reduced fractal dimension), three scaling functions are introduced to unify the responses of different materials. The critical exponents are demonstrated to be universal regardless of the randomness in various constitutive properties and their random noise levels.

This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length scale R. The focus is on pre-fractal media (i.e. those with lower and upper cutoffs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. This procedure allows a specification of a geometry configuration of continua by 'fractal metric' coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D, d and R, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure, making it capable of grasping local fractal anisotropy. Finally, the one-, two-and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.

Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscop-ically uniform loadings. The materials are cubic-shaped domains (of up to 100 Â 100 Â 100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J 2 flow rule. The spatial assignment of the grains' elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The ran-domness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the 'reduced von-Mises stress', 'reduced plastic volume fraction' and 'reduced fractal dimension'), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitu-tive properties and their random noise levels.