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The hydrodynamic theory of active nematics has been often used to describe the spatio-temporal dynamics of cell flows and motile topological defects within soft confluent tissues. Those theories, however, often rely on the assumption that…

Due to their microstructure, quasi brittle materials present rough cracks. Under sliding of the crack lips, this roughness involves in one hand induced opening and in the other hand some apparent plasticity which is due to the interlocking…

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation…

We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory. The complexity of physical phenomena appears as encoded in so called…

We propose a variational phase-field model of fracture capable of accounting for arbitrary closed convex strength domains. Unlike traditional models based on Ambrosio and Tortorelli regularization, the phase-field variable does not affect…

The constitutive behaviors of materials are often modeled using a network of different rheological elements. These rheological models are mostly developed within a one-dimensional small strain framework. One of the key impediments of…

In all structural models, the section or fiber response is a relation between the strain measures and the stress resultants. This relation can only be expressed in a simple analytical form when the material response is linear elastic. For…

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a…

In this third decade of systems engineering in the twenty-first century, it is important to develop and demonstrate practical methods to exploit machine-readable models in the engineering of systems. Substantial investment has been made in…

Identifying the regions responsible for plastic flow in amorphous solids remains an open problem, since structural disorder seems to prevent the direct application of concepts such as dislocations, topological defects that successfully…

One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and…

Under mechanical deformation, most materials exhibit both elastic and fluid (or plastic) responses. No existing formalism derived from microscopic principles encompasses both their fluid-like and solid-like aspects. We define the {\it…

When a single long piece of elastic wire is injected trough channels into a confining two-dimensional cavity, a complex structure of hierarchical loops is formed. In the limit of maximum packing density, these structures are described by…

Cracks are the major vehicle for material failure, and often exhibit rather complex dynamics. The laws that govern their motion have remained an object of constant study for nearly a century. The simplest kind of dynamic crack is a single…

We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras…

We develop a complete theory of non-formal deformation quantization exhibiting a nonzero minimal uncertainty in position. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on…

I propose that Physics should be formulated using minimal mathematical structure, beginning with its foundational arena: spacetime. This paper opens with a concise overview of several research directions explored in previous work. Among…

The deformation of rocks is associated with microcracks nucleation and propagation, i.e. damage. The accumulation of damage and its spatial localization lead to the creation of a macroscale discontinuity, so-called "fault" in geological…

Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in…

A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the…