A mathematical model for plasticity and damage: A discrete calculus formulation

A mathematical model for elasticity using calculus on discrete manifolds

“…or, equivalently, It is easy to observe that for |y i |=|b i |, ie, a 0i =0 or a 1i =0, and from (8), we haveK i ¼ jF i j jb i j . The proof of (b) is trivial, and a similar one can be found in Dassios et al 3 The proof is completed.…”

“…Consider a tessellation of a 3D region into cells, eg, a Voronoi tessellation around a given set of N points, representing a particular material microstructure. 3 The tessellation is a 3-complex denoted by V and containing N 3-cells (cells), N 2 2-cells (faces), N 1 1-cells (edges), and N 0 0-cells (nodes). We explore the possibility to replace V with two 1-complexes (graphs), basic and complimentary, determined by V. The basic graph, denoted by G, has n = N nodes placed at 3-cells' centres, and m = N2 edges connecting nodes whose 3-cells have common 2-cells.…”

“…Lattice models for deformation and fracture of solid materials were developed firstly for quasibrittle materials, such as concretes and rocks, 1,2 and extended recently for elastic-plastic materials, such as structural steels. 3 The benefit of modelling materials, treated as continua in classical mechanics, with discrete lattices is that the nucleation, growth, and coalescence of discontinuities (cracks) become natural processes. As these are nontopological changes in the system, the classical solid mechanics, being a thermodynamic "bulk" theory, does not work.…”

A stability result for a network of two triple junctions on the plane

“…Our interest is in studying a network of curves, which is in motion, with the normal velocity equal to the curvature. For some recent contributions, see [9][10][11][12][13][14][15] and the references therein. A network of two triple junctions can be seen in Figure 1.…”

“…R are real-valued functions of class C 2 , and l i is length of curve Q i . Then from[9,10,14], 8i D 1, 2, 3, 4, 5, we conclude to the following eigenvalue problem…”