The indentation load-displacement behavior of six materials tested with a Berkovich indenter has been carefully documented to establish an improved method for determining hardness and elastic modulus from indentation load-displacement data. The materials included fused silica, soda–lime glass, and single crystals of aluminum, tungsten, quartz, and sapphire. It is shown that the load–displacement curves during unloading in these materials are not linear, even in the initial stages, thereby suggesting that the flat punch approximation used so often in the analysis of unloading data is not entirely adequate. An analysis technique is presented that accounts for the curvature in the unloading data and provides a physically justifiable procedure for determining the depth which should be used in conjunction with the indenter shape function to establish the contact area at peak load. The hardnesses and elastic moduli of the six materials are computed using the analysis procedure and compared with values determined by independent means to assess the accuracy of the method. The results show that with good technique, moduli can be measured to within 5%.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A53D45298E14F73B86495BBC056E16DC/S0884291400017039a.pdf/div-class-title-an-improved-technique-for-determining-hardness-and-elastic-modulus-using-load-and-displacement-sensing-indentation-experiments-div.pdf

An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments

The method we introduced in 1992 for measuring hardness and elastic modulus by instrumented indentation techniques has widely been adopted and used in the characterization of small-scale mechanical behavior. Since its original development, the method has undergone numerous refinements and changes brought about by improvements to testing equipment and techniques as well as from advances in our understanding of the mechanics of elastic–plastic contact. Here, we review our current understanding of the mechanics governing elastic–plastic indentation as they pertain to load and depth-sensing indentation testing of monolithic materials and provide an update of how we now implement the method to make the most accurate mechanical property measurements. The limitations of the method are also discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400085800

Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology

One of the simplest, and most completely treated, fields of application of quantum mechanics is the theory of atoms with one or two electrons For hydrogen and the analcgous ions He+, Li++, etc, the calculations can be performed exactly, both in Schrodinger’s nonrelativistic wave mechanics and in Dirac’s relativistic theory of the electron More specifically, the calculations are exact for a single electron in a fixed Coulomb potential Hydrogen-like atoms thus furnish an excellent way of testing the validity of quantum mechanics For such atoms the correction terms due to the motion and structure of atomic nuclei and due to quantum electrodynamic effects are small and can be calculated with high accuracy Since the energy levels of hydrogen and similar atoms can be investigated experimentally to an astounding degree of accuracy, some accurate tests of the validity of quantum electrodynamics are also possible Finally, the theory of such atoms in an external electric or magnetic field has also been developed in detail and compared with experiment

Quantum Mechanics of One- and Two-Electron Atoms

Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

The mathematical theory of equilibrium cracks in brittle fracture

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy SA = −Tr ρAlogρA corresponding to the reduced density matrix ρA of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

https://iopscience.iop.org/article/10.1088/1742-5468/2004/06/P06002/pdf

Entanglement entropy and quantum field theory

Special functions of mathematical physics and chemistry

Elements of Partial Differential Equations

The stress field in the vicinity of a Griffith crack in a strip of finite width

The distribution of stress in the vicinity of an external crack in an infinite elastic solid

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates (p, φ, z), the displacement vector has only one non-vanishing component uφ (p, z), and the stress tensor has only two non-vanishing components, σρπ(p, z) and σπz(p, z). The stress-strain relations reduce to the two simple equationswhere μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equationis satisfied provided that the function uπ(ρ, z) is a solution of the partial differential equationThe boundary conditions can be written in the formwhere, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, uπ, σρπ and σπz all tend to zero.

/pdf/the-reissner-sagoci-problem-15dier5xh2.pdf

The Reissner-Sagoci problem †

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Ian N. Sneddon

Author Tools

Chat about Author