Visualization of Stress Waves in Elastic Solids

Symbolic and geometric visual abstractions have long been the primary language for communicating mechanical phenomena[1]. For example, the mathematical form of a stress wave, Christoffel's equation, is a tensor relation describing a classical eigenvalue- eigenvector problem. The tensor equation itself is a symbolic abstraction derived from the physical laws of motion and the constitutive relations. The plot of a mathematical form is a tool for deducing the basic trends of a solution. In effect, it is a paradigm of what we suppose the physical solution "looks" like. Understanding of the physical behavior described often requires further mental projection.

The velocity surface (and its variations) is the geometrical abstraction used to represent the solution to Christoffel's equation. Given the velocity surface for a given material, the phase velocity, polarization vector, energy flux deviation and slowness vector can be determined. Knowledge of these parameters allows the scientist to mentally animate a picture of a set of infinite plane waves propagating through the medium of interest. This abstraction is used extensively in geometric solutions to Love (earthquake)waves, Rayliegh waves and reflection-refraction. However, they do not visually simulate an actual wave.

Of course, an actual stress wave is not easily seen since the wavelength is generally smaller than can be discerned by the human eye (excluding the case of an earthquake). Numerical simulation provides less abstract way of visualizing a stress wave solution. Stress waves in a Graphite/Epoxy plate have been numerically simulated using both finite element methods [Kriz and Heyliger][2] and finite difference methods [Kriz and Gary][3]. For this project, I used data sets obtained from both these simulations to create animated visuals of wave morphology.The model consisted of a finite mesh representation of a Graphite/Epoxy plate subject to a sinusoidal boundary condition of finite aperture. Animated sequences of gray scale images were used to observe the deformation fields various values of theta to characterize the total acoustic response of graphite/epoxy.:

Theta: 10, 20, 30, 40, 50, 60 Stress wave theory is developed with the assumption that the waves are of infinite extent and propagate in a semi-infinite medium. The numerical simulation, like any real stress wave, must have bounds, and therefore includes the effects of reflection, diffraction, beam-spreading and surface waves. These effects are easily seen by animating a Gouroud shaded surface representation of the displacement field. In this datamation, the height of the shaded surface is proportional to the total displacement. The angle theta is 50^o and corresponds to a mode transition in the QT and QL waves. This same data set was used to create a SIGgraph98 video ( wave.mov (17Mb) / wave.mpg (24Mb)).

This web summary was created by J. Boyet Stevens, Graduate Student Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia

References:

1. R. B. Haber, " Visualization Techniques for Engineering Mechanics," Computing Systems in Engineering, Vol. 1, No. 1, pp. 37-50 , 1990.

2. R.D. Kriz and J.M. Gary, 'Numerical Simulation and Visualization Models of Stress Wave Propagation Graphite / Epoxy Composites', Review of Progress in Quantitative NDE, Vol. 9, Eds. D.O. Thompson and D.E. Chimenti, Plenum Press, New York, pp. 125-132, 1990.

3. R.D. Kriz and P.R. Heyliger, "Finite Element Model of Stress Wave Topology in Unidirectional Graphite/Epoxy Composites: Wave Velocities and Flux Deviations", Review of Progress in Quantitative NDE, Vol. 8A, Eds. D.O. Thompson and D.E. Chimenti, Plenum Press, New York, pp. 141-148, 1988.

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