SABRE: Stress Analysis By REflection:
A New Paradigm for Determining the Stress Distribution in Solids 

Paul Hibbert
 
Issue XXI - April 20, 2004
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Abstract

A new method of determining the stress distribution in a solid is presented. It is based on the view that the application of a static force to a body is accomplished by innumerable molecular impacts of the force-inducing medium. Each impact produces a stress wave within the body and thus is reflected many times from the boundaries until it is, for all practical purposes, attenuated to insignificance. The stresses resulting from the sudden shock of an earthquake are well known in the discipline of geophysics and it is proposed that the equations relevant to earthquakes may be applied to calculate static stresses. Due to the random nature of the timing of multiple impacts and the random angle of molecular impact at the point of application of a "static" force, each impact will produce a different stress pattern within the body. It is the contention of the author that the traditional (macroscopic) static stress pattern is the accumulation of the effects of the countless waves.

In order to compute the stress distribution a grid is constructed over the area under investigation, similar to the familiar finite element method. Stress waves are modeled by sending out rays from the impacting force corresponding to whether dilatation (compression /tension) or shear (transverse/distortional) waves are induced. As a ray progresses over the grid, the direction and magnitude of the stress is accumulated in each cell that is affected. Each reflection of a ray from a boundary decomposes into a dilatation ray and a transverse ray; thus a tree structure of rays is produced. The effects of all the rays are accumulated in each cell so that the overall distribution may be observed.

History of the Project


While doing a take-home exam for my Master's Degree in Structural Engineering at the University of British Columbia in 1964, I conceived this alternative approach to stress analysis. During the intervening years I attempted to solve the lengthy equations, which were to prove formidable, and to create a FORTRAN program to incorporate these ideas. I was active in the practical implementation of finite element analysis in those early years and several times I approached university professors with my scheme but, probably because of lack of available time on their part, I was advised to take a course in elasticity or to attempt to apply statistics to the molecular interaction phenomenon. I always had encouragement, but obtained nothing more in the way of concrete support. Perhaps I was not able to adequately describe the technique, hence I have decided to present the proposed technique in a more or less structured way and to run the flag up on the Net and see if anyone salutes it. I must admit that my mathematical ability is not up to the level of tensors and I have plugged away at the problem without coming to a conclusion as to whether this is a valid endeavor or not. I would be most grateful for comments of any kind, especially if they point out a conceptual "fatal flaw", which I have been unable to discover. This, of course, would put the matter to rest. Positive comments would be even more welcome.

Whether this approach may ultimately be computationally superior to the finite element method is a matter on which I have no opinion and at the present is irrelevant to me. What is important is the concept of how static stresses arise in a body in the first place and I regard this whole thing as an intellectual exercise. I expect that there may be errors in what I have proposed but if the major premise is correct then the details can be rectified.

Elementary Background

Elementary physics tells us that all materials that are not at absolute zero are composed of molecules or atoms that are in constant, apparently random, agitation. Solids are defined as those bodies that have sufficient intermolecular attraction to constrain the vibrations so the bonds are not broken. Thus, the macroscopic shape of body is maintained, even though the boundaries at the microscopic level are constantly changing. (One may ask the question, "What is the definition of a boundary when no unique line may be drawn to circumscribe the body, i.e. how does one connect the dots of the constantly moving molecules?")

In structural engineering, a point force is always depicted as an arrow with the head at the point of application and is deemed to be analogous to a knife-edge. An ideal knife-edge is a mathematical singularity, but this can never be realized, as it would slice through any body to which it was applied. In reality, a knife-edge is composed of steel molecules configured so that they have a very fine edge but, at the microscopic level, the edge is composed of vibrating molecules. Depending on the amount of force applied to the knife (by some undefined agent), the magnitude of the impacts may be sufficient to break the molecular bonds of the item being sliced. If the body is not cut, then there is an area of very high shear stresses where the geometry of the solid is deformed by the blade. If the blade is sufficiently blunt (i.e., there are a relatively large number of impacting molecules over a broad area), then there is a significant region underneath the blade that is in compression, in addition to the shear stresses near the sides.

The obvious question arises as to how the force to the knife, from an external agent, comes about, and how the force to the external agent from another external agent occurs, and so on, ad infinitum. Those considerations such as gravitation (i.e. acceleration) are beyond the scope of this discussion.

Pulse vs. Wave


In this description of the SABRE method, the terminology of "pulse" is used to distinguish this phenomenon from a wave, even though the principles of the propagation of waves are used in developing the theory. A wave is characterized by the coordinated motion of a great number of particles at the macroscopic level. For example, take the waves induced by dropping a stone in a pond. The initial motion of the water displaced by the stone creates a wave front that moves radially outward from the stone. Many water molecules are involved in this progression. These particles are temporarily raised above the surrounding water ahead of them and with their forward motion fall to create a trough. The particles at the center are temporarily lower than the surrounding level and are forced upward by the hydrostatic pressure. Thus, a series of progressing crests and troughs is formed. The point of this description is to merely accentuate the fact that a wave, of necessity, must result from a statistically significant number of particles so that the concept of hydrostatic pressure is applicable. The motions of the individual molecules are merely "noise" superimposed on the average of the surrounding molecules.

A pulse, on the other hand, and in the context of this description, is associated with a single impact of a single molecule of a solid on another molecule and is considered to be typical of the impacts that constitute the "applied force". In reality, the many impacts that comprise the force vary in velocity (including direction) and occur in random sequence (or, at least, indeterminate, and variable, frequency). The magnitude of the force is only as constant as the statistical average of the impacts is constant. It is because of this random behavior of the multitude of exciting impacts that inhibit the propagation of a true wave. One of the central points of this discussion is that the effect of a concentrated force (in the traditional sense) can be analyzed by observing the effect of an average impact of a single molecule on the stressed body. It is contended that the "static" state of stress of a body is merely the statistical average of the direction and magnitude of the molecular activity at each point and that this is a result of the effects of the pulses generated by the molecules of the applied force. This sets up a reaction that, in any practical sense, is unpredictable, as is the case of many billiard balls ricocheting off each other as a result of a single, initial impact. Such initial impacts take place at intervals in the order of millionths of a second and the timing is irregular so that each pulse may tend to reinforce the formation of a wave, or nullify it. In any case, the result does not exhibit resonant behavior of the body as a whole. The direction and magnitude of the molecular activity are a measure of the stresses that occur in a particular region.

The Bulb of Stress

The bulb of stress is depicted in the below diagram. This theory is most often encountered in the discipline of soil mechanics and defines the stress pattern existing in the soil beneath surface loads, be they concentrated or distributed. This provides the starting point for the SABRE concept. It is postulated that the bulb of stress equation is the solution to the intractable statistical problem of what happens when a multitude of molecules impact the surface molecules of a semi-infinite body at very short, random time intervals. If this is the process, then the bulb of stress equation is the solution.

Bulb of Stress for a Distributed Load
Description of the Algorithm

The effect of the wave is modeled by breaking the wave front into a large number of rays with a constant angle between them, which emanate from the load. The body has a rectangular grid superimposed on it for bookkeeping purposes. Each ray is processed to its completion before processing the next one. The stress (pure dilatational or pure distortional, depending on the type of imposed load) existing on the first path of the ray is computed. This depends on the material properties of the solid, the magnitude of the force and the angle of the ray with the surface. A distance increment from the load is given, the coordinates of the new position are computed and the cell corresponding to this position is identified. The pure stress is converted to its horizontal, vertical and shear stress equivalents, according to Mohr's circle, and they are stored separately for that cell. Distance increments are repetitively applied and the components stored in the appropriate cells until a boundary is encountered. When this occurs, the original ray decomposes into two reflected rays, a dilatational ray and a distortional ray, each with different angles of reflection. The coordinates of the boundary intersection are stored and the magnitudes of the two new stresses, and the angles of the two new rays, are computed. The new ray corresponding to the original will have its sign reversed and have reduced magnitude. The accumulated effect of the incident and reflected rays must balance at the boundary, i.e. there are no applied external forces at the point of reflection.

The reflected dilatational ray is followed next and the process continues by applying distance increments and accumulating the stress components in the cells that were landed on. The dilatational ray is always followed until it has gone as far as necessary for the required attenuation. The coordinates of the last boundary intersection and the magnitude of the distortional stresses are retrieved and the distortional ray is followed until a new boundary is encountered. The resultant dilatational ray is always followed in preference to the distortional ray, if there is any option. In this way, all the branches of the tree structure are followed.
The next ray is processed in the same manner.

When all the rays have been processed, there are three stress components that have been accumulated from a large number of entries for each cell. Each of the cells is processed and converted to principal stresses by means of Mohr's circle. Thus we have determined the stress distribution in the solid. Note, this does not determine the magnitude of the stresses, only the distribution.

The specifying of the distance increment, total number of increments imposed, the number of rays, the cell size, etc. is part of the "art" of the technique, similar to those judgements that must be made when using the finite element method. Proficiency in the art is gained by experience and by calibration against known solutions.

Description of the Computer Program

1. Read in the number of loads, the number of cells in the X-direction and the number of cells in the Y-direction. The loads are assumed to be vertical for this initial version of the program. Even though horizontal components may be read in, the equations are not present to process them.
2. Read the initial random "hit" flag. This controls whether the first "hit" will be at the location of the load or at a random distance along the path.
3. Read in the coordinates of the boundaries. This version of the program is restricted to rectangular boundaries.
4. Read material properties, ray criteria, symmetry conditions i.e.
o distance increment of longitudinal ray
o distance increment of transverse ray
o longitudinal ray length
o symmetry type
o Young's modulus
o Poisson's ratio
5. Compute the ratio of celerities from Poisson's ratio. The celerity of the longitudinal wave is arbitrarily given the value of 1.0.
6. Compute the Lame constant, lambda and shear modulus, G, and the ratio of longitudinal/transverse celerity
7. For this load read its X-coordinate, its Y-coordinate, the magnitude of the horizontal component, the magnitude of the vertical component, the first angle at which a dilatational wave will propagate, the last angle at which a dilatational wave will propagate, and the increment of angle between the first and last rays. The first and last rays would normally be slightly greater than 0 and slightly less than 180 degrees for the straight boundary to which the load is applied (for this primitive version). For testing, only one or two rays would normally be sent out.
8. Compute the initial starting point for the longitudinal ray.
9. The initial "hit" point on each ray is a random number uniformly distributed between 0 and the distance increment when the flag (random indicator) is set to 1, otherwise the first hit point is at the distance increment for the dilatational or distortional wave from the load. The distance increment is computed so that the longitudinal and transverse rays will be sampled the same number of times. The incremental stresses for the cell that the hit occurs in are accumulated in separate variables, which always contain positive entries. If the incremental stress is negative, then the variable is known to contain negative values, even though the value is stored as a positive one. This assists in the Mohr's circle calculations later.
10. Continue incrementing the distance along the ray and accumulating the stresses in the affected cells until a boundary is encountered.
11. When a boundary is encountered, calculate the new stresses for both the dilatational and distortional rays and their angles of reflection, and store those of the distortional ray for future retracing.
12. Follow the dilatational ray to the next boundary, using the same procedure as above and so on until the dilatational ray has gone to completion. Back track by picking up the coordinates of the boundary intersection of the previous distortional ray and its stress, and direction on its new tack.
13. Follow this distortional ray to the next boundary and compute the new components, their stresses and directions.
14. Select the dilatational ray to follow, and so on.
15. Perform the same set of calculations for all of the rays emanating from the source.
16. For each cell, using the components of stored stresses, compute the magnitude and direction of the principal stresses and the shear stresses from Mohr's circle.
17. Compute and display the stress trajectories.

Inverse Square Considerations and Attenuation

The algorithm considers the stresses emanating from a point as existing on a single line and are constant between boundaries. In fact, the true state of affairs is that the stress is reduced in direct ratio with the distance from the source. The diminution of the magnitude of stress with distance is taken care of in the algorithm by the spread of the lines according to the arc increment from ray to ray. As there are deemed to be a sufficiently large number of rays and a sufficiently small distance increment between hits, the inverse linear relationship will be adequately modeled. Because the program is written to consider only plane bodies with a knife edge applying the force over the thickness, the relationship is inverse linear and not inverse square, i.e. we are dealing with only two dimensions.

After only a few reflections the stresses will become very weak and the divergence between successive rays will become great and the rays will span at least several cells. The magnitude of this effect, of course, depends on the initial parameters of cell size, distance increment between hits, and angular increment. When the computation is not terminated, an essentially random process takes place. Stresses continue to be accumulated in random cells, but the magnitudes are very small and the sign of the stresses also becomes random. The net effect after a sufficiently long time is that there is no more accumulation and that further computation is pointless. Thus there is some practical limit of accuracy that can be complied with and this could be automatically computed by reviewing the pattern of the number of hits in neighboring cells. The number of hits should be very large and a smooth pattern should be created.

The Question of Overall Equilibrium

The traditional diagram of a semi-infinite plane with a point force applied to it ignores the problem of equilibrium. It is tacitly assumed, without depicting them, that there are "infinitesimal" forces acting at "infinity" to equilibrate the system. It is the contention of the author that these forces result in a neutral state of stress because of their uniformity and their remoteness. This is not satisfactory for any practical application and all the real forces which exist must be applied. For instance, a simple beam with a load at the center must have the two supporting reactions also applied in the computer simulation.

The Problem of Long, Slender Rods


Consider a long, slender rod under tension or compression. To model this, equal and opposite forces are applied to the ends. Because the rod is slender the stresses are directed longitudinally and when the stress resulting from a central ray is reflected back from an opposite boundary, the sign is reversed, which effectively cancels out the previous value. This applies to both forces. So, apparently there isn't any effective attenuation or spreading of the rays. The number of distance increments is meaningless because all of the stresses from a central ray will cancel except for the portion of the last ray. This is not a refutation of the technique — it merely means that the enormous amount of computation necessary to propagate the shallow rays a sufficiently long time may be impractical.


Bibliography

1. Nadai, A., Plasticity, Engineering Societies Monograph, McGraw-Hill Book Co., 1931, NRC Library QC191 N12.
2. Howell, Benjamin F. Jr., Introduction to Geophysics, McGraw-Hill Book Co. Inc., 1959.
3. Prager, William, Introduction to the Mechanics of Continua, Ginn and Co., 1961, NRC Library QA 901 P7 1961.
4. Kolsky, Herbert, Stress Waves in Solids, Clarendon Press, Oxford, England, 1953. NRC Library QC191 K81
5. Kolsky, Herbert, Stress Waves in Solids, Dover, New York, 1963. NRC Library QC191 K81

Please send comments or questions to:
phibbert@comcast.net

The hand derivations of the programmed equations are available on request. As indicated earlier, they are unsuccessful in obtaining equilibrium at a boundary.

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