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| − | ==1 Title, abstract and keywords<!-- Your document should start with a concise and informative title. Titles are often used in information-retrieval systems. Avoid abbreviations and formulae where possible. Capitalize the first word of the title. | + | == Abstract == |
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| − | Provide a maximum of 6 keywords, and avoiding general and plural terms and multiple concepts (avoid, for example, 'and', 'of'). Be sparing with abbreviations: only abbreviations firmly established in the field should be used. These keywords will be used for indexing purposes.
| + | In this paper, we develop numerical algorithms for the integration of the continuum plastic damage models formulated in the general framework identified in Part I of this work. More specifically, we focus our attention on a particular plastic damage model of porous metals, involving a classical von Mises yield criterion coupled with a pressure dependent damage surface to model the nucleation and growth of voids in the metallic matrix. Unilateral damage leading to a sudden change of stiffness in the material's response due to the closing/opening of these voids is also incorporated through the imposition of the unilateral constraint of a positive void fraction, thus, illustrating the clear physical significance added by this framework in the resulting constitutive models. The proposed integration algorithms fully use the modularity of the identified framework, leading in this way to independent integration algorithms for the elastoplastic part and each damage mechanism. Remarkably, all these individual integration schemes share the same formal structure as the classical return mapping algorithms employed in the numerical integration of elastoplastic models, namely an operator split structure consisting of a trial state and the return map imposing the plastic and damage consistency, respectively. A Newton iterative scheme imposes the equilibrium (equal stresses) among the different mechanisms of the response of the material. This modular structure allows to obtain the closed-form consistent linearization, involving in a simple form the algorithmic consistent tangents corresponding to each independent mechanism, thus resulting in a very modular and efficient computational implementation. The performance of the proposed algorithms is illustrated in several representative numerical simulations. |
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| − | An abstract is required for every document; it should succinctly summarize the reason for the work, the main findings, and the conclusions of the study. Abstract is often presented separately from the article, so it must be able to stand alone. For this reason, references and hyperlinks should be avoided. If references are essential, then cite the author(s) and year(s). Also, non-standard or uncommon abbreviations should be avoided, but if essential they must be defined at their first mention in the abstract itself. -->==
| + | ==Full Document== |
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| − | ==2 The main text<!-- You can enter and format the text of this document by selecting the ‘Edit’ option in the menu at the top of this frame or next to the title of every section of the document. This will give access to the visual editor. Alternatively, you can edit the source of this document (Wiki markup format) by selecting the ‘Edit source’ option.
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| − | 2.1 Subsections
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| − | Divide your article into clearly defined and numbered sections. Subsections should be numbered 1.1, 1.2, etc. and then 1.1.1, 1.1.2, ... Use this numbering also for internal cross-referencing: do not just refer to 'the text'. Any subsection may be given a brief heading. Capitalize the first word of the headings.
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| − | 2.2 General guidelines
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| − | * Follow internationally accepted rules and conventions. In particular use the international system of units (SI). If other quantities are mentioned, give their equivalent in SI.
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| − | 2.3 Tables, figures, lists and equations
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| − | Please insert tables as editable text and not as images. Tables should be placed next to the relevant text in the article. Number tables consecutively in accordance with their appearance in the text and place any table notes below the table body. Be sparing in the use of tables and ensure that the data presented in them do not duplicate results described elsewhere in the article.
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| − | Number the figures according to their sequence in the text. Ensure that each illustration has a caption. A caption should comprise a brief title. Keep text in the illustrations themselves to a minimum but explain all symbols and abbreviations used. Try to keep the resolution of the figures to a minimum of 300 dpi. If a finer resolution is required, the figure can be inserted as supplementary material
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| − | 1. The first entry in this list
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| − | You may choose to number equations for easy referencing. In that case they must be numbered consecutively with Arabic numerals in parentheses on the right hand side of the page. Below is an example of formulae that should be referenced as eq. (1].
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| − | 2.4 Supplementary material
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| − | Supplementary material can be inserted to support and enhance your article. This includes video material, animation sequences, background datasets, computational models, sound clips and more. In order to ensure that your material is directly usable, please provide the files with a preferred maximum size of 50 MB. Please supply a concise and descriptive caption for each file. -->==
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| − | ==3 Bibliography<!--
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| − | ==4 Acknowledgments<!-- Acknowledgments should be inserted at the end of the document, before the references section. -->==
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| − | ==5 References<!--[1] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Article code. Available: http://www.scipedia.com/ucode.
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| − | [2] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Volume number, first page-last page.
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| − | [3] Author, C. (Year). Title of work: Subtitle (edition.). Volume(s). Place of publication: Publisher.
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| − | [4] Author of Part, D. (Year). Title of chapter or part. In A. Editor & B. Editor (Eds.), Title: Subtitle of book (edition, inclusive page numbers). Place of publication: Publisher.
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| − | [5] Author, E. (Year, Month date). Title of the article. In A. Editor, B. Editor, and C. Editor. Title of published proceedings. Paper presented at title of conference, Volume number, first page-last page. Place of publication.
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| − | [6] Institution or author. Title of the document. Year. [Online] (Date consulted: day, month and year). Available: http://www.scipedia.com/document.pdf.
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In this paper, we develop numerical algorithms for the integration of the continuum plastic damage models formulated in the general framework identified in Part I of this work. More specifically, we focus our attention on a particular plastic damage model of porous metals, involving a classical von Mises yield criterion coupled with a pressure dependent damage surface to model the nucleation and growth of voids in the metallic matrix. Unilateral damage leading to a sudden change of stiffness in the material's response due to the closing/opening of these voids is also incorporated through the imposition of the unilateral constraint of a positive void fraction, thus, illustrating the clear physical significance added by this framework in the resulting constitutive models. The proposed integration algorithms fully use the modularity of the identified framework, leading in this way to independent integration algorithms for the elastoplastic part and each damage mechanism. Remarkably, all these individual integration schemes share the same formal structure as the classical return mapping algorithms employed in the numerical integration of elastoplastic models, namely an operator split structure consisting of a trial state and the return map imposing the plastic and damage consistency, respectively. A Newton iterative scheme imposes the equilibrium (equal stresses) among the different mechanisms of the response of the material. This modular structure allows to obtain the closed-form consistent linearization, involving in a simple form the algorithmic consistent tangents corresponding to each independent mechanism, thus resulting in a very modular and efficient computational implementation. The performance of the proposed algorithms is illustrated in several representative numerical simulations.
