iMechanica - configurational forces
https://imechanica.org/taxonomy/term/3085
enRevisiting Nucleation in the Phase-Field Approach to Brittle Fracture
https://imechanica.org/node/24284
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/31">fracture</a></div><div class="field-item odd"><a href="/taxonomy/term/2277">strength</a></div><div class="field-item even"><a href="/taxonomy/term/3085">configurational forces</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Twenty years in since their introduction, it is now plain that the regularized formulations dubbed as phase-field of the variational theory of brittle fracture of Francfort and Marigo (1998) provide a powerful macroscopic theory to describe and predict the propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. Over the past ten years, the ability of the phase-field approach to also possibly describe and predict crack nucleation has been under intense investigation. The first of two objectives of this paper is to establish that the existing phase-field approach to fracture at large — irrespectively of its particular version </span>—<span> cannot possibly model crack nucleation. This is so because it lacks one essential ingredient: the strength of the material.</span></p>
<p><span>The second objective is to amend the phase-field theory in a manner such that it can model crack nucleation, be it from large pre-existing cracks, small pre-existing cracks, smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The central idea is to implicitly account for the presence of the inherent microscopic defects in the material </span>—<span> whose defining macroscopic manifestation is precisely the strength of the material </span>—<span> through the addition of an external driving force in the equation governing the evolution of the phase field. To illustrate the descriptive and predictive capabilities of the proposed theory, the last part of this paper presents sample simulations of experiments spanning the full range of fracture nucleation settings.</span></p>
<p><a href="https://www.sciencedirect.com/science/article/pii/S0022509620302623">https://www.sciencedirect.com/science/article/pii/S0022509620302623</a></p>
</div></div></div>Fri, 12 Jun 2020 15:02:58 +0000Oscar Lopez-Pamies24284 at https://imechanica.orghttps://imechanica.org/node/24284#commentshttps://imechanica.org/crss/node/24284Partial constraint singularities in elastic rods
https://imechanica.org/node/22072
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/8024">elastica</a></div><div class="field-item odd"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item even"><a href="/taxonomy/term/3086">material forces</a></div><div class="field-item odd"><a href="/taxonomy/term/27">adhesion</a></div><div class="field-item even"><a href="/taxonomy/term/706">contact mechanics</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>We present a unified classical treatment of partially constrained elastic rods. Partial constraints often entail singularities in both shapes and reactions. Our approach encompasses both sleeve and adhesion problems, and provides simple and unambiguous derivations of counterintuitive results in the literature. Relationships between reaction forces and moments, geometry, and adhesion energies follow from the balance of energy during quasistatic motion. We also relate our approach to the configurational balance of material momentum and the concept of a driving traction. The theory is generalizable and can be applied to a wide array of contact, adhesion, gripping, and locomotion problems.</span></p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/HannaSinghVirga2017.pdf" type="application/pdf; length=442956" title="HannaSinghVirga2017.pdf">Partial_constraint_singularities_in_elastic_rods</a></span></td><td>432.57 KB</td> </tr>
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</div></div></div>Wed, 24 Jan 2018 02:03:09 +0000Harmeet Singh22072 at https://imechanica.orghttps://imechanica.org/node/22072#commentshttps://imechanica.org/crss/node/22072Torsional locomotion
https://imechanica.org/node/17256
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/5451">torsion beam</a></div><div class="field-item odd"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item even"><a href="/taxonomy/term/7109">propulsion</a></div><div class="field-item odd"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item even"><a href="/taxonomy/term/5088">large deformations</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Can a torque induce longitudinal motion of an elastic rod?</span></p>
<p><span>See the explanation and an example of use of the 'torsional gun' at <a href="http://www.ing.unitn.it/~bigoni/torsional_locomotion.html">http://www.ing.unitn.it/~bigoni/torsional_locomotion.html</a></span></p>
<p><iframe src="//www.youtube.com/embed/3qghYI5Ot24?rel=0" frameborder="0" width="560" height="315"></iframe></p>
<p><em>If you're having trouble playing videos on YouTube, <a href="http://www.ing.unitn.it/~bigoni/torsional_locomotion2.html" target="_blank"><em>click here to watch it.</em></a></em></p>
<p> </p>
<p><span><span>More information about my research activity can be found in <a title="http://www.ing.unitn.it/~bigoni/" href="http://www.ing.unitn.it/%7Ebigoni/">http://www.ing.unitn.it/~bigoni/</a><br /> More information about our experiments can be found in <a title="http://ssmg.ing.unitn.it/" href="http://ssmg.unitn.it/">http://ssmg.unitn.it/</a></span></span></p>
<p><img src="http://erc-instabilities.unitn.it/erc-instabilities-logo.jpg" alt="" width="150" height="106" /></p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/bigoni_dalcorso_misseroni_bosi_torsional_locomotion_PRSA2014.pdf" type="application/pdf; length=624826" title="bigoni_dalcorso_misseroni_bosi_torsional_locomotion_PRSA2014.pdf">Open Access article "Torsional locomotion" published in PRSA (2014)</a></span></td><td>610.18 KB</td> </tr>
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</div></div></div>Mon, 29 Sep 2014 16:04:28 +0000Davide Bigoni17256 at https://imechanica.orghttps://imechanica.org/node/17256#commentshttps://imechanica.org/crss/node/17256Elastica arm scale
https://imechanica.org/node/16997
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/8024">elastica</a></div><div class="field-item even"><a href="/taxonomy/term/472">large deformation</a></div><div class="field-item odd"><a href="/taxonomy/term/3085">configurational forces</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>Can configurational forces be exploited to design a new type of scale?</span></p>
<p><span>See the explanation and an example of use at <a href="http://www.ing.unitn.it/~bigoni/elasticscale.html">http://www.ing.unitn.it/~bigoni/elasticscale.html</a></span></p>
<p><iframe src="//www.youtube.com/embed/c_NamRa3lYg?rel=0" frameborder="0" width="560" height="315"></iframe></p>
<p><em>If you're having trouble playing videos on YouTube, <a href="http://www.ing.unitn.it/~bigoni/elasticscale2.html" target="_blank"><em>click here to watch it.</em></a></em></p>
<p> </p>
<p><span><span>More information about my research activity can be found in <a title="http://www.ing.unitn.it/~bigoni/" href="http://www.ing.unitn.it/%7Ebigoni/">http://www.ing.unitn.it/~bigoni/</a><br /> More information about our experiments can be found in <a title="http://ssmg.ing.unitn.it/" href="http://ssmg.unitn.it/">http://ssmg.unitn.it/</a></span></span></p>
<p><img src="http://erc-instabilities.unitn.it/erc-instabilities-logo.jpg" alt="" width="150" height="106" /></p>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/elastic_scale_bosi_misseroni_dalcorso_bigoni_RSPA2014.pdf" type="application/pdf; length=997159" title="elastic_scale_bosi_misseroni_dalcorso_bigoni_RSPA2014.pdf">Preprint of "An elastica arm scale" published in PRSA (2014)</a></span></td><td>973.79 KB</td> </tr>
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</div></div></div>Fri, 08 Aug 2014 09:33:43 +0000Davide Bigoni16997 at https://imechanica.orghttps://imechanica.org/node/16997#commentshttps://imechanica.org/crss/node/16997Configurational balance in slender bodies
https://imechanica.org/node/8957
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/667">beams</a></div><div class="field-item odd"><a href="/taxonomy/term/3085">configurational forces</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In this paper we propose a new derivation of the evolution equation of a sharp, coherent interface in a two-phase body having elongated shape, a body which we regard as a one-dimensional micropolar continuum. To this aim, we introduce a system of forces acting at the interface, and we apply the method of virtual powers to derive a balance law involving these forces. By exploiting the dissipation inequality, we manage to write this balance law in terms of a scalar field whose form is reminiscent of a well-known expression for the configurational stress in three dimensional micropolar continua.
</p>
<p>
The original paper has been published on the Archive of Applied Mechanics. <span class="doi"><span class="label">DOI:</span> <span class="value">10.1007/s00419-010-0470-3</span></span>
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<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://imechanica.org/files/eshelbeam.pdf" type="application/pdf; length=133527" title="eshelbeam.pdf">eshelbeam.pdf</a></span></td><td>130.4 KB</td> </tr>
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</div></div></div>Thu, 23 Sep 2010 07:38:53 +0000Giuseppe_Tomassetti8957 at https://imechanica.orghttps://imechanica.org/node/8957#commentshttps://imechanica.org/crss/node/8957What is stress? Who has ever seen stress? Is stress a physical quantity?
https://imechanica.org/node/8872
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item odd"><a href="/taxonomy/term/5550">displacement</a></div><div class="field-item even"><a href="/taxonomy/term/5551">invariant integral</a></div><div class="field-item odd"><a href="/taxonomy/term/5552">pseudo physical quantity</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong><span>What is stress? Who has ever seen stress? Is stress a physical quantity?</span></strong> </p>
<p class="MsoNormal">
<span>Professor Yi-Heng Chen, Xi’an Jiaotong University, 710049, P.R.China</span>
</p>
<p class="MsoNormal">
<span>e-mail: <a href="mailto:yhchen2@mail.xjtu.edu.cn">yhchen2@mail.xjtu.edu.cn</a></span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>In fact, this question has been bothering the present author for more than 15 years.</span>
</p>
<p class="MsoNormal">
<span>As well-known, all previous and present researchers who were/are majoring in solid mechanics in mechanical engineering or aerospace engineering always used the classical concept of stress as they used other physical quantities: displacement or strain etc.</span>
</p>
<p class="MsoNormal">
<span>However, there is no one in the world, who has ever seen stress by using any tool!</span>
</p>
<p class="MsoNormal">
<span>Moreover, no one really understood what the physical meaning of stress is!</span>
</p>
<p class="MsoNormal">
<span>Or instead, no one recognized the detailed fact whether stress is a real physical quantity?</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>In is well-known that for a given geometric coordinate system the displacements of a mass point of a solid are actually physical quantities without any doubt! This is because the displacements of a mass point could be directly seen or measured by using eyes, some advanced optic instruments, or even the electron microscope. </span>
</p>
<p class="MsoNormal">
<span>Thus their grads are also physical quantity at the same point where (or around a small region) the continuous first order differentials exist. </span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>The major difficulty is what the stress is?</span>
</p>
<p class="MsoNormal">
<span>Obviously, the concept of stress is quite different from the strain and displacements at least due to the following three viewpoints:</span>
</p>
<p class="MsoNormal">
<span>(1) First, the stress is invisible or un-measurable by using any existing instruments including all optic instruments and electrical instruments, whereas the displacements or strains, as well known, are measurable. For example, some new optic instruments could be used to clarify mere 0.01 micro meter of displacements and 1 micro strain as GOM or other corporations reported. Moreover, some advanced electric microscope has 0.18 nano meter resolving power!<span> </span></span>
</p>
<p class="MsoNormal">
<span>(2) Second, the classical concept of stress is based on the generalized Hook law in elasticity and then it is extended to treat some structural problems in plasticity such as the plastic deformation theory or plastic fluid theory. More recently, this concept is extended to micro mechanics, damage mechanics, even nano mechanics. However, strictly speaking, this concept is not yielded from experimental observations but from the man’s brain! Thus, it is not a real physical quantity or, in other words, it is a pseudo physical quantity, just an imagined physical quantity! This question is easy to be proved because no one in the literature who claimed that he has seen the stress or he has measured the stress! </span>
</p>
<p class="MsoNormal">
<span>(3) Third, in non-linear and inhomogeneous materials under some loadings, there are many crystals (metal) or the particles (ceramic) with the size scale of several micros. Generally speaking, the stress field in an inhomogeneous material is not uniform or even not unique although the strain field or displacement field in the same inhomogeneous material is unique (the strain field could uniquely deduced from the measured displacement field). Due to micro defects nucleation, growth, coalescence etc (a nonreversible thermodynamic process), the measured displacement field varies and the deduced strain field varies as well. But each strain/displacement field might lead to several different stress fields, depending on the different constitutive relations established by researchers (or from researches’ brain). In other words, each constitutive relation would only yield a special stress field. Many constitutive relations would yield many different stress fields but researchers actually don’t see or measure their stress fields. </span>
</p>
<p class="MsoNormal">
<span>The things become clear! </span>
</p>
<p class="MsoNormal">
<span>The stress concept is actually established from the man’s brain rather than established from real observations! </span>
</p>
<p class="MsoNormal">
<span>This is a major obstacle at 21 century in advanced solid mechanics for modern materials because the advanced technology promotes the instruments becoming smaller and smaller such as MEMS or NEMS, and then researchers majoring in solid mechanics face on some challenge to study small scale mechanics such as micro mechanics or nano mechanics. However, no one could tell us whether the stress (as a macroscopic and pseudo physical quantity) concept is still valid in micomechanics or nano mechanics with defects? </span>
</p>
<p class="MsoNormal">
<span>If so, he should tell us as why?</span>
</p>
<p class="MsoNormal">
<span>If not so, he should tell us what is the possible and alternative physical quantity instead of the stress?</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>This question is very clear as shown in the following figures (attached from websites).</span>
</p>
<p class="MsoNormal">
<span>Figure 1 shows a microscope photo. There are many solid particulars in the photo in the inhomogeneous material. The question is what is the stress on the surface of each particular? How large the stress is? Is this valuable to find the detailed distribution of the stress field as the displacement field or strain field? More importantly, from the micro scale viewpoint, whether the imagined stress field, if it exists, could be used to introduce some phenomenological parameters to evaluate the material failure?</span>
</p>
<p class="MsoNormal">
</p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 1. The first example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 2. The second example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Figure 2 also shows a detailed displacement field as well as the strain field but no one could obtain the detailed stress field although the detailed strain-displacement field could be measured.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 3. The third example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Figure 3 show another displacement field with some surface cracks. However, it is still unclear as what is the stress field although the strain-displacement field could be measured. </span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Moreover, some famous experts have claimed that the stress in their analyses at the nano scale is as large as 100GPa! This stress is over the material elastic modulus? This result privates an evidence that the stress concept is not realistic in nano mechanics with defects.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Of course, it is not fair to overthrow the previous contributions based on the so-called stress analyses. Indeed, all classical strength theories including the fracture mechanics were established from the stress analyses which yielded a large amount of fortune and received significant attention from researches!</span>
</p>
<p class="MsoNormal">
<span>Also, the author does not wish to simply throw over this concept. This might upside down! </span>
</p>
<p class="MsoNormal">
<span>The goal of the present paper is to motivate the further discussions with other researchers who are interested in the topic and to introduce an alternative physical quantity to replace the stress concept, especially in inhomogeneous mechanics, micro mechanics and nano mechanics.</span>
</p>
<p class="MsoNormal">
<span>From the physical view point, the configurational force and the associated invariant integrals might be a useful choice instead of working in stress analyses. This is because these concepts are induced directly from Eshelby’s force with defects, which is based on energy balance viewpoint. In fact, the author has some initial attempts at this research direction [1-17] including one book [11] summarizing the potentials applications of the projected conservation laws of Jk-vector and the M-integral and the L-integral.</span>
</p>
<p class="MsoNormal">
<span>Other advances in this topic such as the Fatigue Damage Driving Force (FDDF) for a cloud of micro-defects will be reported in the author’s subsequent papers.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>[1] Chen Yi-Heng., On the contribution of discontinuities in a near-tip stress field to the J-integral, <em>International Journal of Engineering Science</em>, Vol. 34, 819-829(1996).</span>
</p>
<p class="MsoNormal">
<span>[2] Han J. J, and Chen Yi-Heng., On the contribution of a micro-hole in the near-tip stress field to the J-integral, <em>International Journal of Fracture</em>, Vol. 85, 169-183(1997).</span>
</p>
<p class="MsoNormal">
<span>[3] Zhao L. G, and Chen Yi-Heng., On the contribution of subinterface microcracks near the tip of an interface crack to the J-integral in bimaterial solids, <em>International Journal of Engineering Science</em>, Vol. 35, 387-407(1997).</span>
</p>
<p class="MsoNormal">
<span>[4] Chen Yi-Heng., and Zuo Hong, Investigation of macrocrack-microcrack interaction problems in anisotropic elastic solids-Part I. General solution to the problem and application of the J-integral, <em>International Journal of Fracture, </em>Vol. 91, 61-82(1998).</span>
</p>
<p class="MsoNormal">
<span>[5] Chen Yi-Heng., and Han J. J.<span> </span>Macrocrack-microcrack interaction in piezoelectric materials, Part I. Basic formulations and J-analysis, <em>ASME Journal of Applied Mechanics</em>, Vol. 66, No. 2, 514-521 (1999).</span>
</p>
<p><span>[6] Chen Yi-Heng</span><span>.,<span> and Han J.J. Macrocrack-microcrack interaction in piezoelectric materials, Part II. Numerical results and Discussions, <em>ASME Journal of Applied Mechanics</em>, Vol. 66, No. 2, 522-527(1999).</span></span><span>[7] Tian W.Y. and Chen Yi-Heng., A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials, Part I, Fundamental formulations and the J-integral analysis, <em>International Journal of Solids and Structures,</em> Vol. 37, 7717-7730 (2000). </span><span>[8] Chen Yi-Heng., and Tian W.Y. A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials, Part II, Numerical results and discussions, <em>International Journal of Solids and Structures</em>, Vol.37, 7731-7742 (2000). </span><span>[9] Chen Yi-Heng</span><span>., <span>M-integral analysis for two-dimensional solids with strongly interacting cracks, Part I. In an infinite brittle sold, <em>International Journal of Solids and Structures</em>., Vol. 38/18, 3193-3212 (2001). </span></span></p>
<p class="MsoNormal">
<span>[10] Chen Yi-Heng., M-integral analysis for two-dimensional solids with strongly interacting cracks, Part II. In the brittle phase of an infinite metal/ceramic bimaterial, <span> </span><em>International Journal of Solids and Structures.</em>, Vol. 38/18, 3213-3232 (2001).</span>
</p>
<p><span>[11] </span><em><span>Books: Advances in conservation laws and energy release rates, Kluwer Academic Publishers,</span></em><em><span> </span></em><em><span>The Netherlands (ISBN 1402005008)</span></em><em><span>, 2002</span></em><em><span>.</span></em><span>[</span><span>12</span><span>] Chen, Y.H., and Lu, T.J., (2003) </span><span>Recent developments and applications in invariant integrals, </span><em><span>ASME Applied Mechanics Reviews, </span></em><span>Vol. 56, 515-552<em>.</em></span> </p>
<p class="MsoNormal">
<span>[13] Li Q, Chen YH. (2008) Surface effect and size dependence on the energy release due to a<span> </span>nanosized hole expansion in plane elastic materials, <em>ASME Journal Applied Mechanics, </em>Vol. 75, Novermber.</span>
</p>
<p class="MsoNormal">
<span>[14] Hu Y.F., and Chen Yi-Heng, (2009), M-integral description for a strip with two holes before and after coalescence, <em>Acta Mechanica</em>, Vol. 204(1), 109-.</span>
</p>
<p class="MsoNormal">
<span>[15] Hu Y.F., and Chen Yi-Heng, (2009), M-integral description for a strip with two cracks before and after coalescence, <em>ASME Journal Applied Mechaics,</em> Vol.76, November, 061017-1-12.</span>
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<span>[16] Hui T., and Chen Yi-Heng, (2010), The M-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings, ASME Journal of Applied Mechanics, Vol. 77.</span><strong><span> 021019-1-9.</span></strong>
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<span>[17] Hui T., and Chen Yi-Heng, (2010), The two state M-integral for a nano inclusion in plane elastic materials, ASME Journal of Applied Mechanics, Vol. 77.</span><strong><span> 024505-1-5.</span></strong>
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</div></div></div>Fri, 10 Sep 2010 11:44:38 +0000Yi-Heng Chen8872 at https://imechanica.orghttps://imechanica.org/node/8872#commentshttps://imechanica.org/crss/node/8872Journal Club Theme of May 2009: Configurational forces in finite elements - energy-based mesh adaptation
https://imechanica.org/node/5377
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/447">Finite Element Method</a></div><div class="field-item odd"><a href="/taxonomy/term/821">Journal Club Forum</a></div><div class="field-item even"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item odd"><a href="/taxonomy/term/3853">mesh adaptation</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>*This issue of the Journal Club has been prepared as a group by <a href="http://imechanica.org/user/11313">Ankush Aggarwal</a>, <a href="http://imechanica.org/user/13027">Mo Bai</a>, <a href="http://imechanica.org/user/13015">Mainak Sarkar</a>, and <a href="http://imechanica.org/user/11343">Jee E Rim</a>, and with comments and suggestions from <a href="http://imechanica.org/user/1081">Bill Klug</a> . </p>
<p> Most of us are familiar with continuum mechanics of homogeneous solids, and the use of finite elements in these applications. An oft-arising practical issue is that of meshing - the results and sometimes even the tractability of a finite element problem depend on the suitability of the mesh. In particular, for problems with large differences in the deformation gradient, a mesh that adapts to the most optimal configuration is desirable but difficult to obtain. In this issue of the Journal Club, we propose to discuss a method of mesh adaptation - energy-based mesh-adaptivity and it's connection to configurational forces. Though none of us leading this discussion are experts in the field, we think that this is a very interesting field that is useful to several important finite element problems.</p>
<p> The theory of configurational forces can be traced back to the original article by <cite>Eshelby (1951)</cite>, where the concept of forces acting on a singularity was introduced into the classic theory of elasticity by defining the energy-momentum tensor (or the Eshelby stress tensor). In simple terms, the variation of strain energy density in the material due to a defect or inhomogeneity leads to "configurational" forces acting on these inhomogeneities, which are allowed to move through the material. The resulting stress tensor is the superposition of the strain energy density and a transformation of the Cauchy stress. In keeping with the original concept, historically, the configurational or material forces have mainly been used in analyzing physical defects such as interstitial atoms in solids, dislocation lines, interfaces, and cracks. A nice article with various such examples and further references for each is <cite>Gross et al.(2003)</cite></p>
<p> A relatively new application of configurational forces is in the context of finite elements, where <cite>Braun (1997)</cite> recognized that the finite element discretization can be seen as a kind of material inhomogeneity. We're used to thinking of the finite element approximation of the deformation as being a function of the spatial nodal positions. However, the approximation also depends on the global shape functions and their supports, which are determined by the material nodal point positions. Therefore, the discrete potential energy <em>I</em> can be regarded as a function of both <strong><em>x</em></strong> and <strong><em>X</em></strong>, using the standard notation of <strong><em>x</em></strong> for spatial and <strong><em>X</em></strong> for material points of the domain: <em>I</em> = <em>I</em>(<strong><em>x</em></strong>, <strong><em>X</em></strong>). It naturally follows that by minimizing the discrete potential energy with respect to <strong><em>X</em></strong>, or, in other words, enforcing a balance of configurational forces at each material nodal point, an optimal mesh may be found. If this is a little confusing to visualize, we found this article by Braun (recommended article 1) listed below very helpful: it offers a simple one-dimensional example where the configurational forces and their application in mesh optimization can be followed in detail and explicitly in closed-form.</p>
<p> These ideas are quite simple and pleasing, since an optimal mesh defined as that minimizing the total energy using variational principles makes sense physically as well as mathematically. In addition, measures such as error estimates or mesh adaptation indicators are completely unnecessary. However, in practice, the complete variation of energy with respect to discretization is very difficult, as is the numerical implementation of the minimization problem. This is partly due to the fact that unlike a physical defect which may move freely through the material, the movement of material nodes is restricted by the connectivity of the mesh. In particular, the movement of certain material nodes, such as those on the domain boundary, is further restricted by the necessity of maintaining the domain geometry. In addition, the resulting adapted mesh has to be of acceptable quality, <em> i.e.</em>, with positive Jacobians and a small enough distortion, meaning that constraints - either implicit or explicit, need to be introduced. The main issues to be resolved in our opinion therefore lie in the efficient and practical application of the concept of configurational force balance for mesh adaptation. Here, we suggest two articles as examples of how these difficulties have been (partially) addressed, by invoking suitable approximations and solution schemes. Both demonstrate significant improvements in the finite element solutions of crack-growth problems with mesh adaptation.</p>
<p> <strong>Recommended reading:</strong></p>
<ol><li>M. Braun, "Configurational forces in discrete elastic systems", <em>Archive of Applied Mechanics</em>, 2007, <strong>77</strong>:85-93. (<a href="http://dx.doi.org/10.1007/s00419-006-0076-y">http://dx.doi.org/10.1007/s00419-006-0076-y</a>)
<p> We recommend this paper as a starting point. By dealing with a simple 1-D finite-element problem, this paper provides a clear and easy-to-understand explanation of the concept of configurational forces and investigates its role in mesh optimization. The finite-element solution for nodal displacements are explicitly derived as functions of the material nodal positions, and thereby the direct dependence of the discrete potential energy on the material nodal positions. </p></li>
<li>P. Thoutireddy and M. Ortiz, "A variational r-adaption and shape-optimization method for finite-deformation elasticity", <em>International Journal for Numerical Methods in Engineering</em>, 2004, <strong>61</strong>:1-21 (<a href="http://dx.doi.org/10.1002/nme.1052">http://dx.doi.org/10.1002/nme.1052</a>)
<p> The authors of this paper address the problem of movement of material nodes while maintaining the mesh quality by edge-face or octahedral swapping. This allows limited changes in mesh-topology (the total number of nodes is unchanged), so that while the resulting mesh is not likely to be the minimizer of <em>I</em>, the nodes may move closer towards the minimum without the mesh becoming entangled. While effective, this method is admittedly ad hoc. The flexibility of mesh connectivity is also somewhat undermined by the necessity of maintaining the integrity of the domain boundary - the boundary nodes are constrained to stay within the boundary. These issues were addressed further in a later paper by <cite>Mosler and Ortiz (2006) </cite>. </p></li>
<li>M. Scherer, R. Denzer, and P. Steinmann, "Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics", <em>International Journal of Fracture</em>, 2007, <strong>147</strong>:117-132. (<a href="http://dx.doi.org/10.1007/s10704-007-9143-9">http://dx.doi.org/10.1007/s10704-007-9143-9</a>)
<p> Here, in contrast to the above article, the element connectivities are held fixed during the mesh adaptation. Instead, excessive distortion of the mesh is prevented by introducing inequality constraints that dictate an admissible maximum distortion for each element. The measure for the element distortional deformation can be chosen to be twice continuously differentiable, so that the explicit derivatives can easily be implemented in a finite element method. An advantage of this approach is that the element constraints can be made independent of the original element by defining the distortion measure relative to an arbitrary reference element - thus the original quality of the mesh is not important.</p></li>
</ol><p>In general, the simultaneous minimization of the discrete potential energy <em>I</em>(<strong><em>x</em></strong>, <strong><em>X</em></strong>)with respect to both <strong><em>x</em></strong> and <strong><em>X</em></strong> is problematic due to the non-convexity of <em>I</em>. Thus, both articles 2 and 3 employ a staggered scheme (conjugate gradient and Newton, respectively), for the minimization: first, the configurational forces are computed from equilibrated displacement fields, then the material nodes are shifted to satisfy the configurational force balance, and lastly a new mechanical equilibrium is found for the new material nodal positions. Unfortunately, not enough information is presented for a comparison of computational cost between the two strategies. We envision ongoing and future development in strategies for increased robustness and efficiency - as well as investigations into broader classes of problems.</p>
<p> <em>Further References:</em></p>
<ol><li>J. D. Eshelby, "The force on an elastic singularity", <em>Philos. Trans. Roy. Soc. London Ser. A</em>, 1951, <strong>244</strong>:87-112.</li>
<li>D. Gross, S. Kolling, R. Mueller, I. Schmidt, "Configurational forces and their application in solid mechanics", <em>Eur. J. Mech. A</em>, 2003, <strong>22</strong>:669-692.</li>
<li>M. Braun, "Configurational forces induced by finite-element discretization", <em>Proc. Estonian Acad. Sci. Phys. Math</em>, 1997, <strong>46</strong>:24-31.</li>
<li>J. Mosler, M. Ortiz, "On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations", <em>Int. J. Numer. Meth. Engng</em>, 2006, <strong>67</strong>:1272-1289.</li>
</ol></div></div></div>Thu, 30 Apr 2009 19:31:51 +0000Jee E Rim5377 at https://imechanica.orghttps://imechanica.org/node/5377#commentshttps://imechanica.org/crss/node/53774th International Symposium on Defect and Material Mechanics (ISDMM09), TRENTO, July 6th-9th, 2009
https://imechanica.org/node/4339
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/74">conference</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/179">solid mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/420">cracks</a></div><div class="field-item even"><a href="/taxonomy/term/499">dislocations</a></div><div class="field-item odd"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item even"><a href="/taxonomy/term/3086">material forces</a></div><div class="field-item odd"><a href="/taxonomy/term/3087">phase boundaries</a></div><div class="field-item even"><a href="/taxonomy/term/3088">shape optimization</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The Solid Mechanics group of the University of Trento (Italy) announces ISDMM09, the fourth international meeting devoted to Mechanics of Material Forces, following the workshops held at Kaiserslautern (2003), Symi (2005) and Aussois (2007), and will be held in the alpine city of Trento, Italy, from July 6th to 9th, 2009.
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<strong>Abstract submission: January 31, 2009</strong>
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More info: <a href="http://portale.unitn.it/events/isdmm09">http://portale.unitn.it/events/isdmm09</a>
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</div></div></div>Thu, 20 Nov 2008 14:14:21 +0000Massimiliano Gei4339 at https://imechanica.orghttps://imechanica.org/node/4339#commentshttps://imechanica.org/crss/node/4339Error | iMechanica