A Note on Congruences of Infinite Bounded Involution Lattices

We prove that an infinite (bounded) involution lattice and even pseudo-Kleene algebra can have any number of congruences between 2 and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets. Furthermore, when they have at most as many congruences as elements, these involution lattices and even pseudo-Kleene algebras can be chosen such that all their lattice congruences preserve their involutions, so that they have as many congruences as their lattice reducts. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice and even pseudo-Kleene algebra can have any number of congruences between 2 and its number of subsets, regardless of its number of ideals. Consequently, the same holds for antiortholattices, a class of paraorthomodular Brouwer-Zadeh lattices. Regarding the shapes of the congruence lattices of the lattice{ ordered algebras in question, it turns out that, as long as the number of congruences is not strictly larger than the number of elements, they can be isomorphic to any nonsingleton well-ordered set with a largest element of any of those cardinalities, provided its largest element is strictly join-irreducible in the case of bounded lattice-ordered algebras and, in the case of antiortholattices with at least 3 distinct elements, provided that the predecessor of the largest element of that well-ordered set is strictly join{irreducible, as well; of course, various constructions can be applied to these algebras to obtain congruence lattices with different structures without changing the cardinalities in question. We point out sufficient conditions for analogous results to hold in an arbitrary variety.

Analysis of planes within reduced micromorphic model

A plane within reduced micromorphic model subjected to external static load is studied using the finite element method. The reduced micromorphic model is a generalized continuum theory which can be used to capture the interaction of the microstructure. In this approach, the microstructure is homogenized and replaced by a reduced micromorphic material model. Then, avoiding the complexity of the microstructure, the reduced micromorphic model is analyzed to reveal the interaction of the microstructure and the external loading. In this study, the three-dimensional formulation of the reduced micromorphic model is dimensionally reduced to address a plane under in-plane external load. The governing system of partial differential equations with corresponding consistent boundary conditions are discretized and solved using the finite element method. The classical and nonclassical deformation measures are then demonstrated and discussed for the first time for a material employing the reduced micromorphic model.

A new model of bone remodeling and turnover set up in the framework of generalized continuum mechanics

In the past few decades, numerous studies have attempted to address the various phenomena that take place simultaneously during bone remodeling. Drawing from Frost’s “mechanostat theory,” multiple phenomenological models of varying complexity have been developed to describe bone remodeling in terms of evolution of bone porosity, tissue properties, and mineralization. The main goal of this paper is to present the general theory of a novel macroscopic and comprehensive model of bone remodeling accounting for the interactions of mechanics and biochemistry at the microscale. Two independent remodeling mechanisms are considered: the rotation of the material axes and the turnover of bone material. The former mechanism is related to the change of orientation of bone microstructure. Bone turnover refers to the dynamic process of bone resorption by osteoclasts, formation of unmineralized bone by osteoblasts, and mineralization. The model is set up in the framework of generalized continuum mechanics. The evolution of bone tissue is thus described through its macroscopic deformation as well as macroscopic variables related to the orientation of bone microstructure and bone tissue composition (porosity, unmineralized and mineralized bone matrix). Thermodynamically consistent evolution laws of bone material are obtained by enforcing suitable statements of the virtual power principle and of the dissipation principle. Moreover, additional constitutive hypotheses are formulated to develop a phenomenological law of bone turnover. The turnover model is discussed on the basis of a number of numerical simulations. Although the model can capture the main features of bone turnover, it cannot describe satisfactorily the complexity of the underlying biological crosstalk, highlighting the need of a more refined mechanobiological constitutive theory of bone turnover.

BONE ORTHOTROPIC REMODELING AS A THERMODYNAMICALLY-DRIVEN EVOLUTION

In this paper, we present and discuss a model of bone remodeling set up in the framework of the theory of generalized continuum mechanics which was first introduced by DiCarlo et al. [Sur le remodelage des tissus osseux anisotropes, Comptes Rendus Mécanique 334(11):651–661, 2006]. Bone is described as an orthotropic body experiencing remodeling as a rotation of its microstructure. Thus, the complete kinematic description of a material point is provided by its position in space and a rotation tensor describing the orientation of its microstructure. Material motion is driven by energetic considerations, namely by the application of the Clausius–Duhem inequality to the microstructured material. Within this framework of orthotropic remodeling, some key features of the remodeling equilibrium configurations are deduced in the case of homogeneous strain or stress loading conditions. First, it is shown that remodeling equilibrium configurations correspond to energy extrema. Second, stability of the remodeling equilibrium configurations is assessed in terms of the local convexity of the strain and complementary energy functionals hence recovering some classical energy theorems. Eventually, it is shown that the remodeling equilibrium configurations are not only highly dependent on the loading conditions, but also on the material properties.