Field Theory of Multiscale Plasticity

Download Field Theory of Multiscale Plasticity PDF Online Free

Author :
Publisher : Cambridge University Press
ISBN 13 : 1108875203
Total Pages : 861 pages
Book Rating : 4.1/5 (88 download)

DOWNLOAD NOW!


Book Synopsis Field Theory of Multiscale Plasticity by : Tadashi Hasebe

Download or read book Field Theory of Multiscale Plasticity written by Tadashi Hasebe and published by Cambridge University Press. This book was released on 2023-05-31 with total page 861 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique book provides a concise and systematic treatment of foundational material on dislocations and metallurgy and an up-to-date discussion of multiscale modeling of materials, which ultimately leads to the field theory of multiscale plasticity (FTMP). Unlike conventional continuum models, this approach addresses the evolving inhomogeneities induced by deformation, typically as dislocation substructures like dislocation cells, as well as their interplay at more than one scale. This is an impressively visual text with many and varied examples and viewgraphs. In particular, the book presents a feasible constitutive model applicable to crystal plasticity-based finite element method (FEM) simulations. It will be an invaluable resource, accessible to undergraduate and graduate students as well as researchers in mechanical engineering, solid mechanics, applied physics, mathematics, materials science, and technology.

Multiscale Materials Modelling

Download Multiscale Materials Modelling PDF Online Free

Author :
Publisher : Elsevier
ISBN 13 : 184569337X
Total Pages : 307 pages
Book Rating : 4.8/5 (456 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Materials Modelling by : Z. X. Guo

Download or read book Multiscale Materials Modelling written by Z. X. Guo and published by Elsevier. This book was released on 2007-05-31 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiscale materials modelling offers an integrated approach to modelling material behaviour across a range of scales from the electronic, atomic and microstructural up to the component level. As a result, it provides valuable new insights into complex structures and their properties, opening the way to develop new, multi-functional materials together with improved process and product designs. Multiscale materials modelling summarises some of the key techniques and their applications.The various chapters cover the spectrum of scales in modelling methodologies, including electronic structure calculations, mesoscale and continuum modelling. The book covers such themes as dislocation behaviour and plasticity as well as the modelling of structural materials such as metals, polymers and ceramics. With its distinguished editor and international team of contributors, Multiscale materials modelling is a valuable reference for both the modelling community and those in industry wanting to know more about how multiscale materials modelling can help optimise product and process design. - Reviews the principles and applications of mult-scale materials modelling - Covers themes such as dislocation behaviour and plasticity and the modelling of structural materials - Examines the spectrum of scales in modelling methodologies, including electronic structure calculations, mesoscale and continuum modelling

Crystal Plasticity Finite Element Methods

Download Crystal Plasticity Finite Element Methods PDF Online Free

Author :
Publisher : John Wiley & Sons
ISBN 13 : 3527642099
Total Pages : 188 pages
Book Rating : 4.5/5 (276 download)

DOWNLOAD NOW!


Book Synopsis Crystal Plasticity Finite Element Methods by : Franz Roters

Download or read book Crystal Plasticity Finite Element Methods written by Franz Roters and published by John Wiley & Sons. This book was released on 2011-08-04 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written by the leading experts in computational materials science, this handy reference concisely reviews the most important aspects of plasticity modeling: constitutive laws, phase transformations, texture methods, continuum approaches and damage mechanisms. As a result, it provides the knowledge needed to avoid failures in critical systems udner mechanical load. With its various application examples to micro- and macrostructure mechanics, this is an invaluable resource for mechanical engineers as well as for researchers wanting to improve on this method and extend its outreach.

Multiscale Processes of Instability, Deformation and Fracturing in Geomaterials

Download Multiscale Processes of Instability, Deformation and Fracturing in Geomaterials PDF Online Free

Author :
Publisher : Springer Nature
ISBN 13 : 303122213X
Total Pages : 266 pages
Book Rating : 4.0/5 (312 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Processes of Instability, Deformation and Fracturing in Geomaterials by : Elena Pasternak

Download or read book Multiscale Processes of Instability, Deformation and Fracturing in Geomaterials written by Elena Pasternak and published by Springer Nature. This book was released on 2022-12-12 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: · Proceedings of 12th International Workshop on Bifurcation and Degradation in Geomechanics (IWBDG2022) held on 28 November - 1 December 2022 at the University of Western Australia, in Perth, Australia. The book concentrates on deep understanding of the processes of bifurcation and instability in geoengineering systems. The book covers multiscale processes from the scale of crystals to rocks to rock masses. The book considers a wide range of accompanying phenomena from liquefaction to seismicity and landslides. · Topics covered are: I. Localisation and instability in geomaterials II. Fracturing, failure and seismicity III. Deformation processes Intended readership: Universities and Consulting and Research organisations, research students, academics and engineers working in the fields of geomechanics, rock mechanics and geotechnical engineering.

Computational Plasticity

Download Computational Plasticity PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 3642245900
Total Pages : 550 pages
Book Rating : 4.6/5 (422 download)

DOWNLOAD NOW!


Book Synopsis Computational Plasticity by : Mao-Hong Yu

Download or read book Computational Plasticity written by Mao-Hong Yu and published by Springer Science & Business Media. This book was released on 2012-12-02 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt: “Computational Plasticity with Emphasis on the Application of the Unified Strength Theory” explores a new and important branch of computational mechanics and is the third book in a plasticity series published by Springer. The other two are: Generalized Plasticity, Springer: Berlin, 2006; and Structural Plasticity, Springer and Zhejiang University Press: Hangzhou, 2009. This monograph describes the unified strength theory and associated flow rule, the implementation of these basic theories in computational programs, and shows how a series of results can be obtained by using them. The unified strength theory has been implemented in several special nonlinear finite-element programs and commercial Finite Element Codes by individual users and corporations. Many new and interesting findings for beams, plates, underground caves, excavations, strip foundations, circular foundations, slop, underground structures of hydraulic power stations, pumped-storage power stations, underground mining, high-velocity penetration of concrete structures, ancient structures, and rocket components, along with relevant computational results, are presented. This book is intended for graduate students, researchers and engineers working in solid mechanics, engineering and materials science. The theories and methods provided in this book can also be used for other computer codes and different structures. More results can be obtained, which put the potential strength of the material to better use, thus offering material-saving and energy-saving solutions. Mao-Hong Yu is a professor at the Department of Civil Engineering at Xi'an Jiaotong University, Xi'an, China.

Multiscale and Multiphysics Processes in Geomechanics

Download Multiscale and Multiphysics Processes in Geomechanics PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 3642196306
Total Pages : 223 pages
Book Rating : 4.6/5 (421 download)

DOWNLOAD NOW!


Book Synopsis Multiscale and Multiphysics Processes in Geomechanics by : Ronaldo I. Borja

Download or read book Multiscale and Multiphysics Processes in Geomechanics written by Ronaldo I. Borja and published by Springer Science & Business Media. This book was released on 2011-05-10 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: This state-of-the-art book contains all results and papers of the International Workshop on Multiscale and Multiphysics Processes in Geomechanics at Stanford University Campus, June 23–25, 2010.

Multiscale Modelling in Sheet Metal Forming

Download Multiscale Modelling in Sheet Metal Forming PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3319440705
Total Pages : 416 pages
Book Rating : 4.3/5 (194 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Modelling in Sheet Metal Forming by : Dorel Banabic

Download or read book Multiscale Modelling in Sheet Metal Forming written by Dorel Banabic and published by Springer. This book was released on 2016-10-20 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a unified presentation of the research performed in the field of multiscale modelling in sheet metal forming over the course of more than thirty years by the members of six teams from internationally acclaimed universities. The first chapter is devoted to the presentation of some recent phenomenological yield criteria (BBC 2005 and BBC 2008) developed at the CERTETA center from the Technical University of Cluj-Napoca. An overview on the crystallographic texture and plastic anisotropy is presented in Chapter 2. Chapter 3 is dedicated to multiscale modelling of plastic anisotropy. The authors describe a new hierarchical multi-scale framework that allows taking into account the evolution of plastic anisotropy during sheet forming processes. Chapter 4 is focused on modelling the evolution of voids in porous metals with applications to forming limit curves and ductile fracture. The chapter details the steps needed for the development of dissipation functions and Gurson-type models for non-quadratic anisotropic plasticity criteria like BBC 2005 and those based on linear transformations. Chapter 5 describes advanced models for the prediction of forming limit curves developed by the authors. Chapter 6 is devoted to anisotropic damage in elasto-plastic materials with structural defects. Finally, Chapter 7 deals with modelling of the Portevin-Le Chatelier (PLC) effect. This volume contains contributions from leading researchers from the Technical University of Cluj-Napoca, Romania, the Catholic University of Leuven, Belgium, Clausthal University of Technology, Germany, Amirkabir University of Technology, Iran, the University of Bucharest, Romania, and the Institute of Mathematics of the Romanian Academy, Romania. It will prove useful to postgraduate students, researchers and engineers who are interested in the mechanical modeling and numerical simulation of sheet metal forming processes.

Multiscale Modeling of Complex Materials

Download Multiscale Modeling of Complex Materials PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3709118123
Total Pages : 285 pages
Book Rating : 4.7/5 (91 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Modeling of Complex Materials by : Tomasz Sadowski

Download or read book Multiscale Modeling of Complex Materials written by Tomasz Sadowski and published by Springer. This book was released on 2014-10-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this volume deal with materials science, theoretical mechanics and experimental and computational techniques at multiple scales, providing a sound base and a framework for many applications which are hitherto treated in a phenomenological sense. The basic principles are formulated of multiscale modeling strategies towards modern complex multiphase materials subjected to various types of mechanical, thermal loadings and environmental effects. The focus is on problems where mechanics is highly coupled with other concurrent physical phenomena. Attention is also focused on the historical origins of multiscale modeling and foundations of continuum mechanics currently adopted to model non-classical continua with substructure, for which internal length scales play a crucial role.

Multiscale Materials Modeling for Nanomechanics

Download Multiscale Materials Modeling for Nanomechanics PDF Online Free

Author :
Publisher : Springer
ISBN 13 : 3319334808
Total Pages : 554 pages
Book Rating : 4.3/5 (193 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Materials Modeling for Nanomechanics by : Christopher R. Weinberger

Download or read book Multiscale Materials Modeling for Nanomechanics written by Christopher R. Weinberger and published by Springer. This book was released on 2016-08-30 with total page 554 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a unique combination of chapters that together provide a practical introduction to multiscale modeling applied to nanoscale materials mechanics. The goal of this book is to present a balanced treatment of both the theory of the methodology, as well as some practical aspects of conducting the simulations and models. The first half of the book covers some fundamental modeling and simulation techniques ranging from ab-inito methods to the continuum scale. Included in this set of methods are several different concurrent multiscale methods for bridging time and length scales applicable to mechanics at the nanoscale regime. The second half of the book presents a range of case studies from a varied selection of research groups focusing either on a the application of multiscale modeling to a specific nanomaterial, or novel analysis techniques aimed at exploring nanomechanics. Readers are also directed to helpful sites and other resources throughout the book where the simulation codes and methodologies discussed herein can be accessed. Emphasis on the practicality of the detailed techniques is especially felt in the latter half of the book, which is dedicated to specific examples to study nanomechanics and multiscale materials behavior. An instructive avenue for learning how to effectively apply these simulation tools to solve nanomechanics problems is to study previous endeavors. Therefore, each chapter is written by a unique team of experts who have used multiscale materials modeling to solve a practical nanomechanics problem. These chapters provide an extensive picture of the multiscale materials landscape from problem statement through the final results and outlook, providing readers with a roadmap for incorporating these techniques into their own research.

Crystal Plasticity

Download Crystal Plasticity PDF Online Free

Author :
Publisher : MDPI
ISBN 13 : 3036508384
Total Pages : 438 pages
Book Rating : 4.0/5 (365 download)

DOWNLOAD NOW!


Book Synopsis Crystal Plasticity by : Wojciech Polkowski

Download or read book Crystal Plasticity written by Wojciech Polkowski and published by MDPI. This book was released on 2021-04-27 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a collection of 25 original papers (including one review paper) on state-of-the art achievements in the theory and practice of crystals plasticity. The articles cover a wide scope of research on materials behavior subjected to external loadings, starting from atomic-scale simulations, and a new methodological aspect, to experiments on a structure and mechanical response upon a large-scale processing. Thus, a presented contribution of researchers from 18 different countries can be virtually divided into three groups, namely (i) “modelling and simulation”; (ii) “methodological aspects”; and (iii) “experiments on process/structure/properties relationship”. Furthermore, a large variety of materials are investigated including more conventional (steels, copper, titanium, nickel, aluminum, and magnesium alloys) and advanced ones (composites or high entropy alloys). The book should be interested for senior students, researchers and engineers working within discipline of materials science and solid state physics of crystalline materials.

Multiscale Analysis of Deformation and Failure of Materials

Download Multiscale Analysis of Deformation and Failure of Materials PDF Online Free

Author :
Publisher : John Wiley & Sons
ISBN 13 : 111995648X
Total Pages : 510 pages
Book Rating : 4.1/5 (199 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Analysis of Deformation and Failure of Materials by : Jinghong Fan

Download or read book Multiscale Analysis of Deformation and Failure of Materials written by Jinghong Fan and published by John Wiley & Sons. This book was released on 2011-06-28 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting cutting-edge research and development within multiscale modeling techniques and frameworks, Multiscale Analysis of Deformation and Failure of Materials systematically describes the background, principles and methods within this exciting new & interdisciplinary field. The author’s approach emphasizes the principles and methods of atomistic simulation and its transition to the nano and sub-micron scale of a continuum, which is technically important for nanotechnology and biotechnology. He also pays close attention to multiscale analysis across the micro/meso/macroscopy of a continuum, which has a broad scope of applications encompassing different disciplines and practices, and is an essential extension of mesomechanics. Of equal interest to engineers, scientists, academics and students, Multiscale Analysis of Deformation and Failure of Materials is a multidisciplinary text relevant to those working in the areas of materials science, solid and computational mechanics, bioengineering and biomaterials, and aerospace, automotive, civil, and environmental engineering. Provides a deep understanding of multiscale analysis and its implementation Shows in detail how multiscale models can be developed from practical problems and how to use the multiscale methods and software to carry out simulations Discusses two interlinked categories of multiscale analysis; analysis spanning from the atomistic to the micro-continuum scales, and analysis across the micro/meso/macro scale of continuum.

IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials

Download IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials PDF Online Free

Author :
Publisher : Springer Science & Business Media
ISBN 13 : 940170483X
Total Pages : 442 pages
Book Rating : 4.4/5 (17 download)

DOWNLOAD NOW!


Book Synopsis IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials by : S. Ahzi

Download or read book IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials written by S. Ahzi and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this proceeding are a collection of the works presented at the IUTAM symposium-Marrakech 2002 (October 20-25) which brought together scientists from various countries. These papers cover contemporary topics in multiscale modeling and characterization of materials behavior of engineering materials. They were selected to focus on topics related to deformation and failure in metals, alloys, intermetallics and polymers including: experimental techniques, deformation and failure mechanisms, dislocation-based modelling, microscopic-macroscopic averaging schemes, application to forming processes and to phase transformation, localization and failure phenomena, and computational advances. Key areas that are covered by some of the papers include modeling of material deformation at various scales. At the atomistic scale, results from MD simulations pertaining to deformation mechanisms in nano-crystalline materials as well as dislocation-defect interactions are presented. Advances in modeling of deformation in metals using discrete dislocation analyses are also presented, providing an insight into this emerging scientific technique that can be used to model deformation at the microscale. These papers address current engineering problems, including deformation of thin fIlms, dislocation behavior and strength during nanoindentation, strength in metal matrix composites, dislocation-crack interaction, development of textures in polycrystals, and problems involving twining and shape memory behavior. On Behalf of the organizing committee, I would like to thank Professor P.

Multiscale Materials Modeling

Download Multiscale Materials Modeling PDF Online Free

Author :
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110412454
Total Pages : 346 pages
Book Rating : 4.1/5 (14 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Materials Modeling by : Siegfried Schmauder

Download or read book Multiscale Materials Modeling written by Siegfried Schmauder and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-08-22 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents current spatial and temporal multiscaling approaches of materials modeling. Recent results demonstrate the deduction of macroscopic properties at the device and component level by simulating structures and materials sequentially on atomic, micro- and mesostructural scales. The book covers precipitation strengthening and fracture processes in metallic alloys, materials that exhibit ferroelectric and magnetoelectric properties as well as biological, metal-ceramic and polymer composites. The progress which has been achieved documents the current state of art in multiscale materials modelling (MMM) on the route to full multi-scaling. Contents: Part I: Multi-time-scale and multi-length-scale simulations of precipitation and strengthening effects Linking nanoscale and macroscale Multiscale simulations on the coarsening of Cu-rich precipitates in α-Fe using kinetic Monte Carlo, Molecular Dynamics, and Phase-Field simulations Multiscale modeling predictions of age hardening curves in Al-Cu alloys Kinetic Monte Carlo modeling of shear-coupled motion of grain boundaries Product Properties of a two-phase magneto-electric composite Part II: Multiscale simulations of plastic deformation and fracture Niobium/alumina bicrystal interface fracture Atomistically informed crystal plasticity model for body-centred cubic iron FE2AT ・ finite element informed atomistic simulations Multiscale fatigue crack growth modeling for welded stiffened panels Molecular dynamics study on low temperature brittleness in tungsten single crystals Multi scale cellular automata and finite element based model for cold deformation and annealing of a ferritic-pearlitic microstructure Multiscale simulation of the mechanical behavior of nanoparticle-modified polyamide composites Part III: Multiscale simulations of biological and bio-inspired materials, bio-sensors and composites Multiscale Modeling of Nano-Biosensors Finite strain compressive behaviour of CNT/epoxy nanocomposites Peptide・zinc oxide interaction

Multiscale Simulations and Mechanics of Biological Materials

Download Multiscale Simulations and Mechanics of Biological Materials PDF Online Free

Author :
Publisher : John Wiley & Sons
ISBN 13 : 1118402944
Total Pages : 509 pages
Book Rating : 4.1/5 (184 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Simulations and Mechanics of Biological Materials by : Shaofan Li

Download or read book Multiscale Simulations and Mechanics of Biological Materials written by Shaofan Li and published by John Wiley & Sons. This book was released on 2013-03-19 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiscale Simulations and Mechanics of Biological Materials A compilation of recent developments in multiscale simulation and computational biomaterials written by leading specialists in the field Presenting the latest developments in multiscale mechanics and multiscale simulations, and offering a unique viewpoint on multiscale modelling of biological materials, this book outlines the latest developments in computational biological materials from atomistic and molecular scale simulation on DNA, proteins, and nano-particles, to meoscale soft matter modelling of cells, and to macroscale soft tissue and blood vessel, and bone simulations. Traditionally, computational biomaterials researchers come from biological chemistry and biomedical engineering, so this is probably the first edited book to present work from these talented computational mechanics researchers. The book has been written to honor Professor Wing Liu of Northwestern University, USA, who has made pioneering contributions in multiscale simulation and computational biomaterial in specific simulation of drag delivery at atomistic and molecular scale and computational cardiovascular fluid mechanics via immersed finite element method. Key features: Offers a unique interdisciplinary approach to multiscale biomaterial modelling aimed at both accessible introductory and advanced levels Presents a breadth of computational approaches for modelling biological materials across multiple length scales (molecular to whole-tissue scale), including solid and fluid based approaches A companion website for supplementary materials plus links to contributors’ websites (www.wiley.com/go/li/multiscale)

Multiscale Modeling of Heterogenous Materials

Download Multiscale Modeling of Heterogenous Materials PDF Online Free

Author :
Publisher : John Wiley & Sons
ISBN 13 : 1118623509
Total Pages : 275 pages
Book Rating : 4.1/5 (186 download)

DOWNLOAD NOW!


Book Synopsis Multiscale Modeling of Heterogenous Materials by : Oana Cazacu

Download or read book Multiscale Modeling of Heterogenous Materials written by Oana Cazacu and published by John Wiley & Sons. This book was released on 2013-03-01 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: A material's various proprieties is based on its microscopic and nanoscale structures. This book provides an overview of recent advances in computational methods for linking phenomena in systems that span large ranges of time and spatial scales. Particular attention is given to predicting macroscopic properties based on subscale behaviors. Given the book’s extensive coverage of multi-scale methods for modeling both metallic and geologic materials, it will be an invaluable reading for graduate students, scientists, and practitioners alike.

Advances in Heterogeneous Material Mechanics 2008

Download Advances in Heterogeneous Material Mechanics 2008 PDF Online Free

Author :
Publisher : DEStech Publications, Inc
ISBN 13 : 1932078800
Total Pages : 1757 pages
Book Rating : 4.9/5 (32 download)

DOWNLOAD NOW!


Book Synopsis Advances in Heterogeneous Material Mechanics 2008 by : Jinghong Fan

Download or read book Advances in Heterogeneous Material Mechanics 2008 written by Jinghong Fan and published by DEStech Publications, Inc. This book was released on 2008 with total page 1757 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The International Conference on Heterogeneous Material Mechanics (ICHMM) in Huangshan, China, June 3-8, 2008 follows the successful inaugural ICHMM held in ChongQing, China in June, 2004. The ICHMM series is the first international forum that focuses exclusively on various issues related to the behavior of heterogeneous materials in a broad sense. The object of the ICHMM is to present and publicize integrated scientific and engineering approaches to the measurement and modeling of phenomena at the interface of materials science, physics, chemistry, biology, and solid mechanics."--Preface, p. xxxix.

Advances in Heterogeneous Material Mechanics 2011

Download Advances in Heterogeneous Material Mechanics 2011 PDF Online Free

Author :
Publisher : DEStech Publications, Inc
ISBN 13 : 1605950548
Total Pages : 1205 pages
Book Rating : 4.6/5 (59 download)

DOWNLOAD NOW!


Book Synopsis Advances in Heterogeneous Material Mechanics 2011 by : Jinghong Fan

Download or read book Advances in Heterogeneous Material Mechanics 2011 written by Jinghong Fan and published by DEStech Publications, Inc. This book was released on 2011 with total page 1205 pages. Available in PDF, EPUB and Kindle. Book excerpt: