iMechanica - asymptotic methods
https://imechanica.org/taxonomy/term/505
enA multi-scale modeling framework for instabilities of film/substrate systems
https://imechanica.org/node/19039
<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/219">wrinkling</a></div><div class="field-item odd"><a href="/taxonomy/term/3915">postbuckling</a></div><div class="field-item even"><a href="/taxonomy/term/3369">bifurcation</a></div><div class="field-item odd"><a href="/taxonomy/term/421">multiscale</a></div><div class="field-item even"><a href="/taxonomy/term/505">asymptotic methods</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>Spatial pattern formation in stiff thin films on soft substrates is investigated from a multi-scale point of view based on a technique of slowly varying Fourier coefficients. A general macroscopic modeling framework is developed and then a simplified macroscopic model is derived. The model incorporates Asymptotic Numerical Method (ANM) as a robust path-following technique to trace the post-buckling evolution path and to predict secondary bifurcations. The proposed multi-scale finite element framework allows sinusoidal and square checkerboard patterns as well as their bifurcation portraits to be described from a quantitative standpoint. Moreover, it provides an efficient way to compute large-scale instability problems with a significant reduction of computational cost compared to full models.</p>
<p>Fan Xu, Michel Potier-Ferry.</p>
<p><a href="http://dx.doi.org/10.1016/j.jmps.2015.10.003">http://dx.doi.org/10.1016/j.jmps.2015.10.003</a></p>
<p><strong>Keywords</strong>: Wrinkling; Post-buckling; Multi-scale; Fourier series; Path-following technique; Finite element method.</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/Xu%2C%20Potier-Ferry_JMPS_Multi-scale%20modeling%20film-substrate.pdf" type="application/pdf; length=8570660">Xu, Potier-Ferry_JMPS_Multi-scale modeling film-substrate.pdf</a></span></td><td>8.17 MB</td> </tr>
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</div></div></div>Tue, 27 Oct 2015 10:06:05 +0000Fan Xu19039 at https://imechanica.orghttps://imechanica.org/node/19039#commentshttps://imechanica.org/crss/node/19039Instabilities in thin films on hyperelastic substrates by 3D finite elements
https://imechanica.org/node/18469
<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/219">wrinkling</a></div><div class="field-item odd"><a href="/taxonomy/term/3915">postbuckling</a></div><div class="field-item even"><a href="/taxonomy/term/3369">bifurcation</a></div><div class="field-item odd"><a href="/taxonomy/term/4819">Neo-Hookean</a></div><div class="field-item even"><a href="/taxonomy/term/8380">hyper elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/505">asymptotic methods</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>F. Xu, Y. Koutsawa, M. Potier-Ferry, S. Belouettar</p>
<p><a href="http://dx.doi.org/10.1016/j.ijsolstr.2015.06.007">http://dx.doi.org/10.1016/j.ijsolstr.2015.06.007</a></p>
<p> </p>
<p><strong>Abstract:</strong></p>
<p>Spatial pattern formation in thin films on rubberlike compliant substrates is investigated based on a fully nonlinear 3D finite element model, associating nonlinear shell formulation for the film and finite strain hyperelasticity for the substrate. The model incorporates Asymptotic Numerical Method (ANM) as a robust path-following technique to predict a sequence of secondary bifurcations on their post-buckling evolution path. Automatic Differentiation (AD) is employed to improve the ease of the ANM implementation through an operator overloading, which allows one to introduce various potential energy functions of hyperelasticity in a quite simple way. Typical post-buckling patterns include sinusoidal and checkerboard, with possible spatial modulations, localizations and boundary effects. The proposed finite element procedure allows accurately describing these bifurcation portraits by taking into account various finite strain hyperelastic laws from the quantitative standpoint. The occurrence and evolution of 3D instability modes including fold-like patterns will be highlighted. The need of finite strain modeling is also discussed according to the stiffness ratio of Young's modulus.</p>
<p><strong>Keywords:</strong> Wrinkling; Post-buckling; Bifurcation; neo-Hookean hyperelasticity; Path-following technique; Automatic Differentiation.</p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
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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/Xu%20et%20al._2015_Instabilities%20in%20thin%20films%20on%20hyperelastic%20substrates%20by%203D%20finite%20elements_0.pdf" type="application/pdf; length=4718201">Xu et al._2015_Instabilities in thin films on hyperelastic substrates by 3D finite elements.pdf</a></span></td><td>4.5 MB</td> </tr>
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</div></div></div>Thu, 18 Jun 2015 16:29:15 +0000Fan Xu18469 at https://imechanica.orghttps://imechanica.org/node/18469#commentshttps://imechanica.org/crss/node/18469The Finite Element Analysis of Shells - Fundamentals - Second Edition (Springer, 2011)
https://imechanica.org/node/9931
<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/155">structures</a></div><div class="field-item odd"><a href="/taxonomy/term/248">finite element analysis</a></div><div class="field-item even"><a href="/taxonomy/term/505">asymptotic methods</a></div><div class="field-item odd"><a href="/taxonomy/term/972">shell theory</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 class="Apple-style-span"><strong><span>Dominique Chapelle</span></strong><strong><span> and Klaus-Jürgen Bathe</span></strong></span>
</p>
<p>
<span class="Apple-style-span">This book presents a modern continuum mechanics and mathematical framework to study shell physical behaviors, and to formulate and evaluate finite element procedures. With a view towards the synergy that results from physical and mathematical understanding, the book focuses on the fundamentals of shell theories, their mathematical bases and finite element discretizations. The complexity of the physical behaviors of shells is analysed, and the difficulties to obtain uniformly optimal finite element procedures are identified and studied. Some modern finite element methods are presented for linear and nonlinear analyses. </span>
</p>
<p>
<span class="Apple-style-span">In this Second Edition the authors give new developments in the field and - to make the book more complete - more explanations throughout the text, an enlarged section on general variational formulations and new sections on 3D-shell models, dynamic analyses, and triangular elements. <br /></span>
</p>
<p>
<span class="Apple-style-span">The analysis of shells represents one of the most challenging fields in all of mechanics, and encompasses various fundamental and generally applicable components. Specifically, the material presented in this book regarding geometric descriptions, tensors and mixed variational formulations is fundamental and widely applicable also in other areas of mechanics. </span>
</p>
<p>
<span class="Apple-style-span"><a href="http://www.springer.com/materials/mechanics/book/978-3-642-16407-1" title="Springer link">Springer link</a> <br /></span>
</p>
<p>
</p>
<p>
</p>
</div></div></div>Sat, 12 Mar 2011 10:08:11 +0000dominique.chapelle9931 at https://imechanica.orghttps://imechanica.org/node/9931#commentshttps://imechanica.org/crss/node/9931Weakly Nonlinear Theory of Dynamic Fracture
https://imechanica.org/node/7827
<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/86">singularity</a></div><div class="field-item odd"><a href="/taxonomy/term/185">experimental mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/481">Nonlinear elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/505">asymptotic methods</a></div><div class="field-item even"><a href="/taxonomy/term/685">dynamic fracture</a></div><div class="field-item odd"><a href="/taxonomy/term/810">finite deformation</a></div><div class="field-item even"><a href="/taxonomy/term/995">instability</a></div><div class="field-item odd"><a href="/taxonomy/term/2801">crack tip</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 class="MsoNormal">
A fundamental understanding of the dynamics of brittle fracture remains a challenge of great importance for various scientific disciplines. From a theoretical point of view, a major difficulty in making progress in this problem stems from the fact that it intrinsically involves the coupling between widely separated time and length scales. Brittle fracture is ultimately driven by the release of linear elastic energy stored on large scales, while this energy is being dissipated in the very small scales near the front of a crack, where large stresses and deformations are concentrated and material separation is actually taking place. The strongly nonlinear and dissipative dynamics in the near vicinity of a crack's front controls the rate of crack growth and its direction, and hence its resolution seems relevant. Indeed, there are indications that various fast fracture instabilities (micro-branching, oscillations) are intimately related to the small scale physics near a crack's front. On the other hand, the phenomenology of brittle fracture appears to be rather universal, being qualitatively and quantitatively similar in materials with wholly different micro-structures and dissipative mechanisms (e.g. glassy polymers, structural glasses and elastomer gels).
</p>
<p> </p>
<p class="MsoNormal">
These observations may suggest that a well-established theory of brittle fracture should incorporate a lengthscale that is associated with the near crack front region, but should otherwise be independent of the details of the small scales physics (unless one aims at calculating the fracture energy, i.e. the amount of energy needed to propagate a crack, instead of using it an a phenomenological material parameter). The canonical theory of fracture, linear elastic fracture mechanics (LEFM), is a scale-free theory and hence every lengthscale that appears in this framework is necessarily of a geometrical nature. This immediately implies that the identification of a non-geometric lengthscale entails the extension of LEFM when it breaks down near the front of a crack. As LEFM is based on a linear elastic constitutive behavior, which is only a first term in a more general displacement-gradients expansion, it is expected to break down near the front of a crack, where deformations become large enough to invalidate the linearity assumption. Progress in understanding the physics of this critical, near-front, nonlinear region has been, on the whole, limited by our lack of hard data describing the detailed physical processes that occur within. Due to the microscopic size and near–sound speed propagation of this region, it is generally experimentally intractable.
</p>
<p> </p>
<p class="MsoNormal">
Recently, this experimental barrier was overcome by using a quasi-2D brittle neo-Hookean material (polyacrylamide<span> </span>gel) in which the fracture phenomenology mirrors that of more standard brittle amorphous materials (e.g. soda-lime glass and Plexiglass), but in which the near-tip (a crack-front becomes a crack-tip in 2D) region is significantly larger and moves significantly more slowly [1, 4]. The latter property allows unprecedented, direct and precise measurements of the near-tip fields of rapid cracks. These experiments revealed in detail how the canonical 1/√r fields and parabolic crack tip opening displacement (CTOD) of LEFM break down as the tip is approached.
</p>
<p> </p>
<p class="MsoNormal">
To account for these observations, a weakly nonlinear theory of dynamic fracture was developed based on a systematic displacement-gradients expansion [2, 3]. The theory predicts novel, universal, 1/r singular displacement-gradients and log(r) displacements. It was shown to be in excellent quantitative agreement with the direct near-tip measurements of rapid cracks [2, 3]. The theory also resolves various puzzles in LEFM, such as the fact that the normal (to the crack propagation direction) component of the linear strain tensor ahead of a running crack becomes negative at sufficiently high speeds, which is physically unintuitive [2]. <span> </span>The presence of linear and weakly nonlinear terms in the crack tip solution allows the definition of a new lengthscale (basically by taking the ratio of these terms), that is shown to be related to a high-speed crack tip oscillatory instability [5]. This lengthscale may hold the key for unlocking various open questions in dynamic fracture. The special mathematical properties of the 1/r singularity (which is strictly forbidden in LEFM) and its relation to the concept of autonomy are discussed in detail in [3]. It is important to note that the weakly nonlinear theory is universally applicable since elastic nonlinearities must precede any irreversible behavior as the crack tip is approached.
</p>
<p> </p>
<p class="MsoNormal">
A very recent combined experimental and theoretical study of the large deformation crack-tip region in a neo-Hookean brittle material revealed a hierarchy of linear and nonlinear elastic zones through which energy is transported before being dissipated at a crack’s tip [4]. This result provides a comprehensive picture of how remotely applied forces drive brittle failure and highlight the emergence of a lengthscale associated with nonlinear elastic effects, which are expected to precede near-tip dissipation. The results are corroborated by unprecedented direct measurements of the linear and nonlinear J-integral for cracks approaching the Rayleigh wave speed.<span> </span>
</p>
<p> </p>
<p class="MsoNormal">
It is important to stress that LEFM works perfectly well for the gels used in the experiments where it should - not too close to the tip. In fact, these experiments provide the most comprehensive validation of LEFM under fully dynamic conditions, as this material has successfully “passed” every “test” that LEFM can throw its way (e.g. functional form of fields not too close to the tip, equations of motions in both an infinite medium and strip – soon to be published in Physical Review Letters); therefore, these results can be considered to be much more general than simply relevant for this class of neo-Hookean materials. Finally, these experiments have also demonstrated that – at least in this class of materials – nonlinear effects entirely dominate the behavior of the fields surrounding the crack’s tip. Dissipation may still be considered “point-like” – but this material shows that the two qualitatively different mechanisms for the breakdown of LEFM (nonlinear elasticity and dissipation) are separated. It remains to be seen if this is also characteristic for other materials. Currently, these materials are the only ones for which we have such detailed data.
</p>
<p class="MsoNormal">
</p>
<p class="MsoNormal">
[1] A. Livne, E. Bouchbinder, J. Fineberg,
</p>
<p class="MsoNormal">
<span> </span><strong>Breakdown of Linear Elastic Fracture Mechanics near the Tip </strong>
</p>
<p class="MsoNormal">
<strong> of a Rapid Crack</strong>,
</p>
<p class="MsoNormal">
<a href="http://link.aps.org/doi/10.1103/PhysRevLett.101.264301" target="_blank">Phys. </a><a href="http://link.aps.org/doi/10.1103/PhysRevLett.101.264301" target="_blank">Rev. Lett. 101, 264301 (2008).</a>
</p>
<p class="MsoNormal">
Also: <span> </span><a href="http://arxiv.org/abs/0807.4866" target="_blank">ArXiv:0807.4866</a>
</p>
<p class="MsoNormal">
[2] A. Livne, E. Bouchbinder, J. Fineberg, <strong>Weakly Nonlinear Theory </strong>
</p>
<p class="MsoNormal">
<strong> of Dynamic Frcature</strong><br /><span> </span><a href="http://link.aps.org/doi/10.1103/PhysRevLett.101.264302" target="_blank">Phys. Rev. Lett. 101, 264302 (2008).</a>
</p>
<p class="MsoNormal">
Also: <a href="http://arxiv.org/abs/0807.4868" target="_blank">ArXiv:0807.4868</a>
</p>
<p class="MsoNormal">
[3] E. Bouchbinder, A. Livne, J. Fineberg, <br /><span> </span><span> </span><strong>The 1/r Singularity in Weakly Nonlinear Fracture Mechanics</strong> <br /><span> </span><a href="http://dx.doi.org/10.1016/j.jmps.2009.05.006" target="_blank">J. Mech. Phys. Solids 57, 1568 (2009).</a>
</p>
<p class="MsoNormal">
Also: <a href="http://arxiv.org/abs/0902.2121" target="_blank">ArXiv:0902.2121</a>
</p>
<p class="MsoNormal">
[4] A. Livne, E. Bouchbinder, I. Svetlizky, J. Fineberg,
</p>
<p class="MsoNormal">
<span> </span><strong>The Near-Tip Fields of Fast Cracks<br /></strong><span> </span><a href="http://www.sciencemag.org/cgi/content/abstract/327/5971/1359" target="_blank">Science 327, 1359 (2010).</a>
</p>
<p class="MsoNormal">
[5]<span> </span>E. Bouchbinder
</p>
<p class="MsoNormal">
<span> </span><strong>Dynamic Crack Tip Equation of Motion: High-speed </strong>
</p>
<p class="MsoNormal">
<strong> Oscillatory Instability</strong>
</p>
<p class="MsoNormal">
<a href="http://link.aps.org/doi/10.1103/PhysRevLett.103.164301" target="_blank">Phys. Rev. Lett. 103, 164301 (2009).</a>
</p>
<p class="MsoNormal">
Also: <a href="http://arxiv.org/abs/0908.1178" target="_blank">ArXiv:0908.1178</a>
</p>
<p> </p>
</div></div></div>Wed, 17 Mar 2010 15:40:11 +0000Eran Bouchbinder7827 at https://imechanica.orghttps://imechanica.org/node/7827#commentshttps://imechanica.org/crss/node/7827ASYMPTOTIC ELASTIC STRESS FIELDS AT SINGULAR POINTS
https://imechanica.org/node/676
<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/505">asymptotic methods</a></div><div class="field-item even"><a href="/taxonomy/term/506">singular points</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>Singular elastic stress fields are generally developed at sharp re-entrant corners and at the end of bonded interfaces between dissimilar elastic materials. This behaviour can present difficulties in both analytical and numerical solution of such problems. For example, excessive mesh refinement might be needed in a finite element solution.
</p><p> Williams (1952) pioneered a method for determining the strength of the dominant singularity by expressing the local field as an asymptotic expansion. The same method has since been used for a variety of situations leading to singular points, including bonded dissimilar wedges and frictionless or frictional contact between bodies with sharp corners. </p>
<p> Information about the strength of the singularity can be used in analytical solutions to choose an appropriate representation for the fields (for example in the choice of quadrature to use in an integral equation formulation of the problem). It can also be used in numerical solutions to suggest the most appropriate form of graded mesh refinement into the corner, or (better) to develop special corner elements with the analytically determined form. However, asymptotic analysis is seldom the primary purpose of such research and it is tempting just to use a large number of elements in the corner and hope for the best. </p>
<p> We have recently developed a Matlab tool for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity, using Williams' method. The basic mathematics for this procedure is given in Section 11.2 of J.R.Barber, <a href="http://www-personal.umich.edu/~jbarber/elasticity/book.html"> <em>Elasticity,</em></a> Kluwer, Dordrecht 2nd edn. (2002), 410pp. However, it is not necessary to have any detailed knowledge of the method in order to use the tool. </p>
<p> A more detailed description of the procedure, including detailed instructions for using the analytical tool has been published in the Journal of Strain Analysis for Engineering Design and can be downloaded <a href="http://www-personal.umich.edu/~jbarber/donghee.pdf"> here.</a> </p>
<p> The user is prompted to input the local geometry of the system, the material properties and the boundary conditions (and interface conditions in the case of composite bodies or problems involving contact between two or more bodies). The tool then computes the dominant eigenvalue and provides as output the equations defining the singular stress and displacement fields and contour plots of these fields. No knowledge of the asymptotic analysis procedure is required of the user. </p>
<p> The tool is written in the software code MATLAB v7.0 with the MATLAB GUI development environment (GUIDE) v2.5 and the MATLAB Symbolic Toolbox v3.1. It provides a graphic interface in which users can define their problem, determine the order of the corresponding singularity and generate the distribution of stress and displacement. Final results are provided in both text and graphic format. </p>
<p> To download the source code, click on <a href="http://www-personal.umich.edu/~donghl/ws/ws.zip">this link </a> and unzip the downloaded file. If and only if you have difficulty opening this link, try <a href="http://www-personal.umich.edu/~jbarber/asymptotics/ws.zip">this one </a>. After downloading and unzipping the file, open MATLAB and start the program with the command `ws'. </p>
<p> Two example programs are included: `williams.wat' and `bogy.wat', which solve the problems of the single wedge and the bi-material wedge respectively. </p>
<p> Please report any problems with the software or any suggestions for additional features or improvements to <a href="mailto:donghl@umich.edu,%20jbarber@umich.edu">Donghee Lee and J.R.Barber</a> </p>
<p> </p>
</div></div></div>Thu, 11 Jan 2007 19:35:50 +0000Jim Barber676 at https://imechanica.orghttps://imechanica.org/node/676#commentshttps://imechanica.org/crss/node/676