We provide an overview of the basic concepts of scaling and dimensional analysis, followed by a review of some of the recent work on applying these concepts to modeling instrumented indentation measurements. Specifically, we examine conical and pyramidal indentation in elastic-plastic solids with power-law work-hardening, in power-law creep solids, and in linear viscoelastic materials. We show that the scaling approach to indentation modeling provides new insights into several basic questions in instrumented indentation, including, what information is contained in the indentation load-displacement curves? How does hardness depend on the mechanical properties and indenter geometry? What are the factors determining piling-up and sinking-in of surface profiles around indents? Can stress-strain relationships be obtained from indentation load-displacement curves? How to measure time dependent mechanical properties from indentation? How to detect or confirm indentation size effects? The scaling approach also helps organize knowledge and provides a framework for bridging micro- and macroscales. We hope that this review will accomplish two purposes: (1) introducing the basic concepts of scaling and dimensional analysis to materials scientists and engineers, and (2) providing a better understanding of instrumented indentation measurements.

Scaling, dimensional analysis, and indentation measurements

High-performance electrode materials are the key to advances in the areas of energy conversion and storage (e.g., fuel cells and batteries). In this Review, recent progress in the synthesis and electrochemical application of transition metal carbides (TMCs) and nitrides (TMNs) for energy storage and conversion is summarized. Their electrochemical properties in Li-ion and Na-ion batteries as well as in supercapacitors, and electrocatalytic reactions (oxygen evolution and reduction reactions, and hydrogen evolution reaction) are discussed in association with their crystal structure/morphology/composition. Advantages and benefits of nanostructuring (e.g., 2D MXenes) are highlighted. Prospects of future research trends in rational design of high-performance TMCs and TMNs electrodes are provided at the end.

/pdf/transition-metal-carbides-and-nitrides-in-energy-storage-and-33qhnc07u1.pdf

Transition Metal Carbides and Nitrides in Energy Storage and Conversion

Microstructural parameters from X-ray diffraction peak broadening

A nanoindentation study of serrated flow in bulk metallic glasses

http://ece.uprm.edu/~msuarez/4001/doc/read_assign3_2ndSem0910.pdf

Nanoindentation studies of materials

In this paper we construct a Floer-homology invariant for a natural and wide class of sutured manifolds that we call balanced. This generalizes the Heegaard Floer hat theory of closed three-manifolds and links. Our invariant is unchanged under product decompositions and is zero for nontaut sutured manifolds. As an application, an invariant of Seifert surfaces is given and is computed in a few interesting cases.

/pdf/holomorphic-discs-and-sutured-manifolds-4gsiwkve9h.pdf

Holomorphic discs and sutured manifolds

Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M,γ)⇝(M′,γ′) is a sutured manifold decomposition then SFH(M′,γ′) is a direct summand of SFH(M,γ). To prove the decomposition formula we give an algorithm that computes SFH(M,γ) from a balanced diagram defining (M,γ) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M,γ) is taut then SFH(M,γ)≠0. Other applications include simple proofs of a result of Ozsvath and Szabo that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry. Moreover, using these methods we show that if K is a genus g knot in a rational homology 3–sphere Y whose Alexander polynomial has leading coefficient ag≠0 and if rkHFK(Y,K,g)<4 then Y∖N(K) admits a depth ≤2 taut foliation transversal to ∂N(K).

/pdf/floer-homology-and-surface-decompositions-38xo5dw240.pdf

Floer homology and surface decompositions

Cobordisms of sutured manifolds and the functoriality of link Floer homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. 
The proof relies on finding a simple generating set for the fundamental group of the "space of Heegaard diagrams," and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

Naturality and mapping class groups in Heegaard Floer homology

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning. A framework through which machine learning can guide mathematicians in discovering new conjectures and theorems is presented and shown to yield mathematical insight on important open problems in different areas of pure mathematics.

András Juhász

Papers

Holomorphic discs and sutured manifolds

Floer homology and surface decompositions

Cobordisms of sutured manifolds and the functoriality of link Floer homology

Naturality and mapping class groups in Heegaard Floer homology

Advancing mathematics by guiding human intuition with AI.