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| | ==Abstract== | | ==Abstract== |
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| − | 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.
| + | This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. |
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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.
| + | The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. |
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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.
| + | Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice. |
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| − | ==2 The main text==
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| + | <pdf>Media:Draft_Samper_728720684_2173_M147.pdf</pdf> |
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| + | ==References== |
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| − | ===2.1 Subsections===
| + | See pdf document |
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| − | ===2.2 General guidelines===
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| − | ===2.3 Tables, figures, lists and equations===
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| − | <span id='_Ref382560620'></span>
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| − | {| style="margin: 1em auto 1em auto;border: 1pt solid black;border-collapse: collapse;"
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| − | | style="text-align: center;"|Thickness
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| − | | style="text-align: center;"|3.175 mm
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| − | | style="text-align: center;"|Young Modulus
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| − | | style="text-align: center;"|12.74 MPa
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| − | | style="text-align: center;"|Poisson coefficient
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| − | | style="text-align: center;"|0.25
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| − | | style="text-align: center;"|Density
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| − | | style="text-align: center;"|1107 kg/m<sup>3</sup>
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| − | |}
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| − | <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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| − | <span style="text-align: center; font-size: 75%;">Table 1: Material properties</span></div>
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| − | <span id='_Ref448852946'></span>
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| − | [[Image:Scipedia.gif|center|480px]]
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| − | <span style="text-align: center; font-size: 75%;">Figure 1. Scipedia logo.</span></div>
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| − | 1. The first entry in this list
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| − | 2. The second entry
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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. <span id='cite-_Ref424030152'></span>[[#_Ref424030152|(1)]].
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| − | | style="vertical-align: top;"| <math>{\nabla }^{2}\phi =0</math>
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| − | | style="text-align: right;"|<span id='_Ref424030152'></span>
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| − | (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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| − | <span id='_Ref449344604'></span>
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| − | Citations in text will follow a citation-sequence system (i.e. sources are numbered by order of reference so that the first reference cited in the document is [<span id='cite-1'></span>[[#1|1]]], the second [<span id='cite-2'></span>[[#2|2]]], and so on) with the number of the reference in square brackets. Once a source has been cited, the same number is used in all subsequent references. If the numbers are not in a continuous sequence, use commas (with no spaces) between numbers. If you have more than two numbers in a continuous sequence, use the first and last number of the sequence joined by a hyphen (e.g. [<span id='cite-1'></span>[[#1|1]], <span id='cite-3'></span>[[#3|3]]] or [<span id='cite-2'></span>[[#2|2]]-<span id='cite-2'></span>[[#4|4]]]).
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| − | <span id='_Ref449084254'></span>
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| − | ==4 Acknowledgments==
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| − | Acknowledgments should be inserted at the end of the document, before the references section.
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| − | ==5 References==
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| − | [[#cite-1|[1]]] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Article code. Available: [http://www.scipedia.com/ucode. http://www.scipedia.com/ucode.]
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| − | [[#cite-2|[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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| − | [[#cite-4|[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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| − | [[#cite-5|[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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This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices.
The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology.
Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice.
