In the general case of force existing in solid elastic materials, it exists in three directions and these are invariably orthogonal. Stress is defined as force per unit area, where area is taken normal to force.
The physical truth or the foregoing can be demonstrated from photoelastic measurements in three dimensions by using "stress freezing." For the two dimensional case, bonded resistance strain gauges are used in either the 45° or 60° configuration to determine the principal strains and their directions. Those strains, and the values of the modulus of elasticity and Poisson's ratio for the material, then provide two stresses.
As will be shown here, the text THEORY OF ELASTICITY by Timoshenko and Goodier is at variance with the foregoing. It is self-evident that the concept of stress"stress components" consisting of "normal and shear stresses" is in valid because the only stresses defined and demonstrated above. Also, since there is no"shear stress," it is self-evident that there is no "modulus of elasticity in shear."
With the respect to the non-existence of a "shear stress," this should have been obvious to Timoshenko sincein his article 12. Construction of Mohr Strain Circle for Strain Rosette, he wrote of Mohr's circle for determining the principal strains from three gauge strain rosettes. Further, in hes article 10. Strain of a point., he effectively defined"shear strain" as "the decrease in angle of a right angle," from which "shear strain" is simply an aspect of deformation, and since all aspects deformation can be derived from the principal strains and geometr, and since the principal strains are due to the stresses, the modulus of elasticity and Poisson's reatio, then as stated in the preceding paragraph, the concept of"stress components" consistingof "normal and shear stresses"is invalid.
The latter points were either not obvious to Timoshenko, or he chose to ignore them because he had already accepted a theory by Neuman and a theory by Poisson. the latter two theories are given in the following pages.
Under Timoshenko's article 2. Stress, F. E. Neuman presented The Theory of Stress Components which was supposed to prove/show the existence of "normal stresses,""shear stresses," and distinctive"principal stresses." The theory purports to show that "the forces acting on a small area in a body under stress can be " reduced to a resultant," and that "the stress corresponding to the resultant," can be "resolved"into "normal" and "shearing" components."
Nemann made his first mistake with his first statement viz., "the forces actin on a small area in a body under stress can be "reduced to a resultant.""Not true. The belief that force"acts on an area" makes the assumption that a geometric plane in a body under stress can contribute to an effect on forces in the body, and thus, that a geometric plane has physical substance, But wevster defined "plane" as, "a two-dimentional extent of zero curvature," and we emphasize that is two-dimentional, not three-dimentional. Thus, a plane has zero thickness, and has no effect on anything. In short, Neumann's theory was invalid, and his error was an error of physics.
The second theory is attributed to S.D. Poisson and is The Theory of a Shearing Strain, a Shearing Stress, and a Modulus of Elasticity in Shear. Poisson's theory is the same as Neumann's with the addition of the so-called" Thus, on p8 of Theory of Elasticity, under his Fig. 7 we find the following: "Let us consider the particular case of deformation of the rectangular parallelopiped in which σy = - σz and σx = 0. Cutting out an element abcd by planes parallel to the x-axis at 45 deg. to the y- and z-axes, it may be seen from Fig,7b (not shown here) by summing up the forces along and perpendicular to bc-" etc. What Poisson overlooked was that as consequences of cutting out the element, the remaining portion of the body was distorted, the area of the hole was greater that the area of the element, and the entire stress field was altered such that σy and σz no longer existed as before. Instead, the latter were replaced by σp and σq where those values changed from point to point and with relative distance from the edges of the hole. Poisson's error was an error of physics, as was Neumann's theory before. In our view it was a shortcoming of Timoshenko's that he was too trusting in so far as he accepted the work of others as a valid without testing that work for errors in physics , logic and mathematics. Also, he lacked that sixth sense which prompts one to recognize that a belief is erroneous. Summary: In the gereral case of force existing in solid elastic materials, it exists in three directions and these are invariably orthogonal. Stress is defined as force per unit area, where area is taken normal to force. Neumann's Theroy of Stress Components was invalid because it required that a geometric plane has physical substance, but Webster showed that a plane is two-dimentional, and from this it follows that it has no phusical substance. Poisson's theory of a Modulus of Elasticity in Shear was invalid because when and elemen is cut out-- among others-- the entire stress field is altered. Timoshenko unintentionally misled the world of Civil and Mechanical Engineering by teaching that the theories by Neumann and Poison were valed, whereas, both theories are invalid.FINIS
Correspondence in respect to the foregoing should be addressed as follows:
J.G. Willmot, P.Eng.
1748 Pagehurst Ave.
Mississauga, ON L4X 1Y2
Canada
