iMechanica - hamilton system
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enStroh formalism and hamilton system for 2D anisotropic elastic
https://imechanica.org/node/3671
<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/669">Stroh formalism</a></div><div class="field-item odd"><a href="/taxonomy/term/670">anisotropic elasticity</a></div><div class="field-item even"><a href="/taxonomy/term/2720">hamilton system</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>
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<span>We have read some papers of stroh formalism and the textbook of Tom Ting, and found that the stroh formalism and the hamilton system proposed by prof.zhong wanxie had some relation. We want to know whether the stroh formalism is enough for the analysis of the anisotropic elastic? Thus's to say, for some problems could not give the satisfied answer which we may try the hamilton framework. I briefly compare the two methods as follows:</span>
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<span>On one hand, I noted that the stroh formalism is widely used in the anisotropic elastic, both static and wave propagation. Stroh formalism is powerful and elegant. In some sense, the stroh formalism may be regarded as the generalization of the complex function method for 2D isotropic elastic, this formalism reveals simple structures hidden in the equations of anisotropic elasticity and provides a systematic approach to these equations.</span>
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<span>On another hand, Prof.zhong wanxie had established a new system for elastic under the hamilton framework. For the isotropic elastic, we can derive the exact solution which is usually in the series solution, without assumptions of the solutions, for the strip domain and sectorial domain via the new system. The key idea of this new system is the introdution of the dual variables---stresses; then one direction is modelled as the time coordinate and using the method of separation variables, we can get the </span><span>eigenvalue problem of Hamiltonian matrix</span><span>: H*phi=miu*phi, where H is a operator matrix and also hamiltonian matrix, next we can get eigenfunction of lamda for the other direction. We found that the ratios lamda/miu has the same meaning of the eigenvalue p; finally, we established the eigenfunction via induced the boundary condition. </span><span> </span>
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<span>Most the solutions having been obtained are isotropic elastic, however, this system can be extened to anisotropic elastic too. Unfortunately, this extention is invalid for sectorial domain, since, the solutions of sectorial domain need to do coordinates transformation which are only found for the situation that the elastic constants are rotation invariant.</span>
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<span>Compare the two methods, the stroh formalism give the general solution and the final solution for some problems need to be determined by the boundary condition; while the hamilton system give the final solution directly in the series solution without the assumption of the solution. However, the hamilton system has a restriction in the shape--strip domain, it may be overcome this by the conformal tranformation, but, in that case, the solution may become so complex that it loses the advantage of the closed-form solution. </span>
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<span>Thank you for your attention, any comments is appreciated.</span><span> </span>
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<thead><tr><th>Attachment</th><th>Size</th> </tr></thead>
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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/Hamiltonian%20system%20based%20Saint%20Venant%20solutions%20for%20multi-layered%20composite%20plane%20anisotropic%20plates.pdf" type="application/pdf; length=132840" title="Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates.pdf">Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates.pdf</a></span></td><td>129.73 KB</td> </tr>
<tr class="even"><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/Plane%20elasticity%20in%20sectorial%20domain%20and%20the%20Hamiltonian%20system%20.pdf" type="application/pdf; length=482125" title="Plane elasticity in sectorial domain and the Hamiltonian system .pdf">Plane elasticity in sectorial domain and the Hamiltonian system .pdf</a></span></td><td>470.83 KB</td> </tr>
<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/A%20state%20space%20formalism%20for%20anisotropic%20elasticity.%20Part%20I%20Rectilinear%20anisotropy%20.pdf" type="application/pdf; length=134677" title="A state space formalism for anisotropic elasticity. Part I Rectilinear anisotropy .pdf">A state space formalism for anisotropic elasticity. Part I Rectilinear anisotropy .pdf</a></span></td><td>131.52 KB</td> </tr>
<tr class="even"><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/A%20state%20space%20formalism%20for%20anisotropic%20elasticity.%20Part%20II%20Cylindrical%20anisotropy%20.pdf" type="application/pdf; length=166593" title="A state space formalism for anisotropic elasticity. Part II Cylindrical anisotropy .pdf">A state space formalism for anisotropic elasticity. Part II Cylindrical anisotropy .pdf</a></span></td><td>162.69 KB</td> </tr>
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</div></div></div>Sun, 10 Aug 2008 13:28:29 +0000Teng zhang3671 at https://imechanica.orghttps://imechanica.org/node/3671#commentshttps://imechanica.org/crss/node/3671