Engineer Energy Dissipation in 3D Graphene Nanolattice Via Reversible Snap-Through Instability

Carbon micro/nanolattice materials, defined as three-dimensional (3D) architected metamaterials made of micro/nanoscale carbon constituents, have demonstrated exceptional mechanical properties, including ultrahigh specific strength, stiffness, and extensive deformability through experiments and simulations. The ductility of these carbon micro/nanolattices is also important for robust performance. In this work, we present a novel design of using reversible snap-through instability to engineer energy dissipation in 3D graphene nanolattices. Inspired by the shell structure of flexible straws, we construct a type of graphene counterpart via topological design and demonstrate its associated snap-through instability through molecular dynamics (MD) simulations. One-dimensional (1D) straw-like carbon nanotube (SCNT) and 3D graphene nanolattices are constructed from a unit cell. These graphene nanolattices possess multiple stable states and are elastically reconfigurable. A theoretical model of the 1D bi-stable element chain is adopted to understand the collective deformation behavior of the nanolattice. Reversible pseudoplastic behavior with a finite hysteresis loop is predicted and further validated via MD. Enhanced by these novel energy dissipation mechanisms, the 3D graphene nanolattice shows good tolerance of crack-like flaws and is predicted to approach a specific energy dissipation of 233 kJ/kg in a loading cycle with no permanent damage (one order higher than the energy absorbed by carbon steel at failure, 16 kJ/kg). This study provides a novel mechanism for 3D carbon nanolattice to dissipate energy with no accumulative damage and improve resistance to fracture, broadening the promising application of 3D carbon in energy absorption and programmable materials.

Homomesy in Products of Three Chains and Multidimensional Recombination

J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, we prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to n dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result does not fully generalize to an arbitrary product of three chains, nor to larger products of chains; however, we have a partial generalization to an arbitrary product of three chains. Additional corollaries include refined homomesy results in the product of three chains and a new result on increasing tableaux. We conclude with a generalization of recombination to any ranked poset and a homomesy result for the Type B minuscule poset cross a two element chain.

A Numerical Study of the Induced Stresses in the Separation Points of the Tensile Element (Chain) of the Plate Conveyor Used in the Blowing Unit in Water Factories

The Induced Stresses in Tensile Element Joints (chain) were Studied in Two Phases: The first phase is to conduct a design study of the plate conveyor to determine the maximum tensile strength to which the joint is exposed. Then build two models, the first one represents a single joint with its components (wedge- copper ring- plate) with the basic dimensions and measurements of the chain. The second model was designed with new dimensions to suit the conveyor's working conditions. In the second phase, the three-dimension finite elements method was used to identify the stresses induced in the joint for both models and then compare the results to identify the model that shows the best performance. The result showed that increasing the external thickness of the joint by double in the proposed model up to the value of 6 mm was able to provide a homogeneous distribution of the main induced stress, which contributed to reducing the critical values of these stresses compared to the induced stresses in the model currently used. Consequently, increasing the external thickness of the joint has played an important role in reducing stresses, which leads to an increase the service life of the plate conveyor chain.

Perfect non-commuting graphs of matrices over chains

In this paper, we study the non-commuting graph [Formula: see text] of strictly upper triangular [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We prove that [Formula: see text] is a compact graph. From [Formula: see text], we construct a poset [Formula: see text]. We further prove that [Formula: see text] is a dismantlable lattice and its zero-divisor graph is isomorphic to [Formula: see text]. Lastly, we prove that [Formula: see text] is a perfect graph.

Zero-divisor graphs of matrices over lattices

In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.

Algebraic functions in Łukasiewicz implication algebras

In this article we study algebraic functions in [Formula: see text]-subreducts of MV-algebras, also known as Łukasiewicz implication algebras. A function is algebraic on an algebra [Formula: see text] if it is definable by a conjunction of equations on [Formula: see text]. We fully characterize algebraic functions on every Łukasiewicz implication algebra belonging to a finitely generated variety. The main tool to accomplish this is a factorization result describing algebraic functions in a subproduct in terms of the algebraic functions of the factors. We prove a global representation theorem for finite Łukasiewicz implication algebras which extends a similar one already known for Tarski algebras. This result together with the knowledge of algebraic functions allowed us to give a partial description of the lattice of classes axiomatized by sentences of the form [Formula: see text] within the variety generated by the 3-element chain.

Iterative Properties of Birational Rowmotion II: Rectangles and Triangles

Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left( p, q\right) $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.

The Virtual Constraint Model and its Application to Tolerance Analysis

The geometrical element chain method (GECM) is a tolerance analysis approach based on parametrical driving theory, with the principle of functional dimension driving being its key theore-tical basis. Currently, the functional dimension driving theory cannot cover the implicit constraints and the geometrical constraints in the 3D models, thus the application of GECM has been limited. To solve this problem, this paper presented the concept of virtual constraint model (VCM) of the pro-ducts 3D models, and constructed the algorithm of tolerance analysis oriented to VCMs to extend the application scope of GECM. Firstly, the mapping matrix between functional dimension set and the modeling parameter set was constructed, and a necessary and sufficient condition of the existence of the mapping relation was given. Secondly, the concept of VCM was put forward, and the tolerance analysis algorithm of the VCM was constructed. Finally, an example was given to illustrate the validity and effectiveness of the approach in this paper.