Thermomechanical theory of layered viscoelastic piezoelectric shells polarized in one coordinate direction

ABSTRACTConcentration of a diffusing substance in a medium was derived in various cases of uni-dimensional diffusion, including in a semi-infinite medium and a plate-shaped medium. Multi-dimensional diffusion involves boundary conditions in each coordinate direction. The algorithm dealing with uni-dimensional case becomes very complicated in multi-dimensional cases. This study proposes an algorithm, which is called the complementary method, that combines complementary functions of the normalized solution in uni-dimensional diffusion case by multiplication to solve those in various multi-dimensional diffusion cases with dramatically simplified mathematics. Besides, the complementary method is used to solve various kinds of boundary conditions for multi-dimensional diffusion.

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Diffractographic Techniques for Vibration Measurement

Journal of Engineering for Industry ◽

10.1115/1.3438364 ◽

1974 ◽

Vol 96 (2) ◽

pp. 553-556 ◽

Cited By ~ 1

Author(s):

W. J. Pastorius ◽

T. R. Pryor

Keyword(s):

Measurement Technique ◽

Reference Point ◽

Vibration Measurement ◽

Coordinate Direction ◽

Diffraction Phenomenon

The single slit diffraction phenomenon is shown to be useful as a noncontacting vibration measurement technique. The diffracting slit is formed between the vibrating object and stationary reference point. Little or no mass need be added to the object. Several readout techniques are described as well as a method for determining vibratory data along a line, in which case vibratory amplitude may be determined at all points in one coordinate direction.

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Thermomechanical theory of materials undergoing large elastic and viscoplastic deformation (AWBA development program)

10.2172/6808737 ◽

1980 ◽

Author(s):

S.E. Martin ◽

J.B. Newman

Keyword(s):

Development Program ◽

Viscoplastic Deformation ◽

Thermomechanical Theory

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Turbulent Boundary-Layer Shear Flows

Fluid Physics in Geology ◽

10.1093/oso/9780195077018.003.0019 ◽

1997 ◽

Author(s):

David Jon Furbish

Keyword(s):

Eddy Viscosity ◽

Shear Flows ◽

Stokes Equations ◽

Streamwise Velocity ◽

Stream Channel ◽

Newtonian Viscosity ◽

Mean Velocity ◽

Navier Stokes Equations ◽

Coordinate Direction ◽

The Mean

Turbulent shear flows next to solid boundaries are one of the most important types of flow in geology. In such flows, turbulence is generated primarily by boundary effects; vorticity originates near a boundary in association with the velocity gradients that arise from the no-slip condition at the boundary. Such gradients provide a ready source of vorticity for eddies and eddy-like structures to develop in response to the destabilizing effects of inertial forces, and then move outward into the adjacent flow. Eddies are also generated within the wakes of bumps that comprise boundary roughness, for example, sediment particles on the bed of a stream channel (Example Problem 11.4.2). As we have seen in Chapter 14, the fluctuating motions of turbulence involve, over any elementary area, fluxes of fluid momentum that are manifest as apparent (Reynolds) stresses. In addition, the complex motions of eddies and eddy-like structures efficiently advect heat and solutes from one place to another within a turbulent flow, and thereby facilitate mixing of heat and solutes throughout the fluid. For similar reasons, turbulent motions are responsible for lofting fine sediment into the fluid column of a stream channel and in the atmosphere. We will concentrate in this chapter on steady unidirectional flows where the mean streamwise velocity varies only in the coordinate direction normal to a boundary and the mean velocity normal to the boundary is zero. We also will adopt a classic treatment of turbulent boundary flow in developing the idea of L. Prandtl’s mixing-length hypothesis, from which we will obtain the logarithmic velocity law, a function that describes how the mean streamwise velocity varies in the coordinate direction normal to a boundary. In developing Prandtl’s hypothesis, we will see that the presence of apparent stresses associated with fluctuating motions leads to the idea of an eddy viscosity or apparent viscosity. Unlike the Newtonian viscosity, the eddy viscosity is a function of the mean velocity, and therefore coordinate position. This means that the eddy viscosity cannot, in general, be removed from stress terms involving spatial derivatives, as we previously did with the Newtonian viscosity in simplifying the Navier–Stokes equations.

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