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Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity. (English) Zbl 1153.74006
Summary: Stroh’s sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium/motion is written as a system of first-order differential equations for the displacement and traction in terms of one of spatial variables. The so-called fundamental elasticity matrix \(N\) appearing in this formulation has the property that, when partitioned as a \(2\times 2\) block matrix, its 12- and 21-blocks are symmetric matrices and its 11-block is the transpose of its 22-block. This property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of Stroh formalism in solving a large variety of problems in general anisotropic elasticity. First, we show that the matrix \(N\) is guaranteed to have the above property by the fact that Stroh formulation is in fact a Hamiltonian formulation with one of the spatial variables acting as the time-like variable. This interpretation provides a much desired guide in dealing with other problems for which the governing equations are different, such as incompressible elasticity and problems associated with anisotropic elastic plates as described by Kirchhoff plate theory. We show that for the last two problems the Hamiltonian interpretation simplifies the derivations significantly, leading to a Stroh formulation in each case which is equivalent to, but much simpler than, what is available in the existing literature.

MSC:
74B05 Classical linear elasticity
74E10 Anisotropy in solid mechanics
74K20 Plates
74J15 Surface waves in solid mechanics
70H05 Hamilton’s equations
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