Generalized stroh formalism for anisotropic elasticity for general boundary conditions

1992 ◽  
Vol 8 (3) ◽  
pp. 193-207 ◽  
Author(s):  
T. C. T. Ting ◽  
Wang Minzhong
Keyword(s):  
Boundary Conditions ◽  
Anisotropic Elasticity ◽  
Stroh Formalism ◽  
General Boundary ◽  
General Boundary Conditions

Generalization of the Stroh Formalism

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Boundary Conditions ◽  
Half Space ◽  
Normal Component ◽  
Stroh Formalism ◽  
General Boundary ◽  
Surface Traction ◽  
General Boundary Conditions ◽  
Special Cases ◽  
Slippery Surface ◽  
Order Stress

It was shown in Chapter 12 that the Stroh formalism can be extended to elastodynamics when the problem is a steady state motion. In this chapter we will see several areas where the Stroh formalism can be generalized. First, the formalism is extended to a group of general boundary conditions that include as special cases the boundary conditions for a traction-free, a rigid, and a slippery surface. Explicit solutions are obtained for the Green's function and the image force in a half-space with a slippery boundary surface subjected to a line dislocation in the half-space. The Stroh formalism is then extended to thermo-anisotropic elasticity where we point out that applications to the interfacial crack may lead to a new and higher order stress singularity. Generalization of the formalism to piezoelectric materials results in an octet formalism for which there are four pairs of complex eigenvalues. Extensions of the Stroh formalism to three-dimensional deformations of anisotropic body require a special attention and are investigated in the next chapter. The Stroh formalism is in terms of the displacement u and the stress function ϕ, both 3-vectors. It is therefore most suitable for boundary conditions that are in terms of the displacement or the surface traction. For a slippery surface the normal component of the displacement and the two tangential components of the surface traction vanish. The boundary conditions are not in terms of u or ϕ, but a mixture of one component from u and two components from ϕ. This destroys the elegant sextic formalism of Stroh. We present here a generalized formalism that is applicable to a class of general boundary conditions. They include as special cases the boundary conditions for a traction-free, a rigid, and a slippery surface. The material presented below is taken from Ting and Wang (1992).

Flexural Vibration Response of Beams With General Boundary Conditions: A Transfer Matrix Approach

1991 ◽  
Author(s):  
T. Önsay
Keyword(s):  
Boundary Conditions ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Response ◽  
General Boundary ◽  
Phase Variable ◽  
General Boundary Conditions ◽  
Transfer Matrices ◽  
Analytical Results

Abstract The wave-mode representation is utilized to obtain a more efficient form to the conventional transfer matrix method for bending vibrations of beams. The proposed improvement is based on a phase-variable canonical state representation of the equation governing the time-harmonic flexural vibrations of a beam. Transfer matrices are obtained for external forces, step-change of beam properties, intermediate supports and for boundaries. The transfer matrices are utilized to obtain the vibration response of a point-excited single-span beam with general boundary conditions. The general characteristic equation and the transfer mobility of a single-span beam are determined. The application of the analytical results are demonstrated on physical structures with different boundary conditions. A hybrid model is developed to incorporate measured impedance of nonideal boundaries into the transfer matrix method. The analytical results are found to be in excellent agreement with experimental measurements.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

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