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Stroh formalism and hamilton system for 2D anisotropic elastic

Teng zhang's picture

 

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:

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.

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 eigenvalue problem of Hamiltonian matrix: 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.  

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.

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.

Thank you for your attention, any comments is appreciated.

Comments

Perhaps you may want to post a preprint or paper of some sort for the details.  Otherwise, it will be hard for others to follow what you really mean.

 Thanks.

Teng zhang's picture

Dear Lim

Thank you for your reminding.

I have added four papers in the post. The first is hamilton system for strip domain; while the second is about the sector domain. The last two papers, written by JQ Tarn, are also in the state space composed by displacements and stress. Tarn's solutions are not in the framework of hamilton, but like the stroh formalism.

 

In contrast to the traditional solution methodology, for which the main technique is semi-inverse
approach, the solutions bases on the hamilton system is carried out rationally in the symplectic space. We want to konw if some problems of anisotropic elastic have not solved well due to the complex assumptions of the solutions, and since, hamilton system need not assumptions of the solutions, it may be help in that kind problems.

 

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