Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Many phenomena in nature, engineering or society when seen at an intermediate distance, in space or time, exhibit the remarkable property of self-similarity: they reproduce themselves as scales change, subject to so-called scaling laws. It's crucial to know the details of these laws, so that mathematical models can be properly formulated and analysed, and the phenomena in question can be more deeply understood.
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We'd like to understand how you use our websites in order to improve them. Register your interest. The main focus of the paper is on similarity methods in application to solid mechanics and author's personal development of Barenblatt's scaling approaches in solid mechanics and nanomechanics. It is argued that scaling in nanomechanics and solid mechanics should not be restricted to just the equivalence of dimensionless parameters characterizing the problem under consideration.
Many of the techniques discussed were introduced by Professor G. Since the author was incredibly lucky to have many possibilities to discuss various questions related to scaling during personal meetings with G. Barenblatt in Moscow, Cambridge, Berkeley and at various international conferences as well as by exchanging letters and electronic mails. Here some results of these discussions are described and various scaling techniques are demonstrated.
The Barenblatt- Botvina model of damage accumulation is reformulated as a formal statistical self-similarity of arrays of discrete points and applied to describe discrete contact between uneven layers of multilayer stacks and wear of carbon-based coatings having roughness at nanoscale.
Another question under consideration is mathematical fractals and scaling of fractal measures with application to fracture. Finally it is discussed the concept of parametrichomogeneity that based on the use of group of discrete coordinate dilation. The parametric-homogeneous functions include the fractal Weierstrass-Mandelbrot and smooth log-periodic functions.
It is argued that the Liesegang rings are an example of a parametric-homogeneous set. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Barenblatt, G. Google Scholar. Willis, J. Solids , , vol. Marigo, J. Lurie, S. Fracture , , vol. USA , , vol. Borodich, F. Earth Science , , vol. Paris , Ser. Mathematical Background, Acta Mech. Some Applications, Acta Mech.
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Reprints and Permissions. Phys Mesomech 22, 73—82 Download citation. Received : 15 November Revised : 15 November Accepted : 22 November Published : 06 April Issue Date : January Search SpringerLink Search. Abstract The main focus of the paper is on similarity methods in application to solid mechanics and author's personal development of Barenblatt's scaling approaches in solid mechanics and nanomechanics.
References 1. Borodich Authors F. Borodich View author publications. You can also search for this author in PubMed Google Scholar. In memory of Professor Grigory Isaakovich Barenblatt.
Rights and permissions Reprints and Permissions. About this article. Cite this article Borodich, F.
Development of Barenblatt’s Scaling Approaches in Solid Mechanics and Nanomechanics
Barenblatt is Emeritus G. Scaling power-type laws reveal the fundamental property of the phenomena--self similarity. The property of self-similarity simplifies substantially the mathematical modeling of phenomena and its analysis--experimental, analytical and computational. The book begins from a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena. Classical examples of scaling phenomena are presented. It is explained why the dimensional analysis as a rule is insufficient for establishing self-similarity and constructing scaling variables. Important examples of scaling phenomena for which the dimensional analysis is insufficient self-similarities of the second kind are presented and discussed.
Susan Mineka , born and raised in Ithaca, New York, received her undergraduate degree magna cum laude in psychology at Cornell University. She received a PhD in xperimental psychology from the University f Pennsylvania, and later completed a formal clinical retraining program from Since she has been Professor of Psychology at Northwestern and since she has served as Director of Clinical Training there. She has taught a wide range of undergraduate and graduate courses, including introductory psychology, learning, motivation, abnormal psychology, and cognitive-behavior therapy.
Barenblatt is Emeritus G. Grigory Isaakovich Barenblatt. Many phenomena in nature, engineering or society when seen at an intermediate distance, in space or time, exhibit the remarkable property of self-similarity: they reproduce themselves as scales change, subject to so-called scaling laws. It's crucial to know the details of these laws, so that mathematical models can be properly formulated and analysed, and the phenomena in question can be more deeply understood. In this book, the author describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity, which are here given a modern treatment. He demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural attributes of self-similarity and shows how and when these notions and tools can be used to tackle the task at hand, and when they cannot. Based on courses taught to undergraduate and graduate students, the book can also be used for self-study by biologists, chemists, astronomers, engineers and geoscientists.