Hysteresis Analysis

by: Howard L. Schriefer, BSME, MSE, PE

 

Hysteresis literally means "belated". In science, hysteresis is the mark of irreversibility because hysteresis results in a return process which differs from the original process. A reversible process is similar to driving from Baltimore, MD to Washington, DC and back on I-95; while an irreversible process would be similar to driving from Baltimore, MD to Washington, DC on I-95, and returning via US 1. Expansion and contraction processes in science are often referred to as reversible or irreversible. This analysis compares processes involving expansion and contraction. Specifically, we are comparing the expansion and subsequent contraction to the contraction and subsequent expansion of a given general sphere having a dimensionless (no assigned units) radius.

 

Before we begin computing, let us recall from calculus that the change in volume of a sphere with respect to its radius is the derivative of the spherical volume with respect to its radius. Interestingly, this derivative is numerically equal to the current spherical surface of the spherical volume. Since the general sphere we herein consider is dimensionless, we can add or subtract the spherical surface to/from the spherical volume. We want to change the spherical volume by one spherical surface because this is the smallest increment for change. In fact, any change less than one spherical surface would render the sphere "unspherical". Intuitively speaking, adding or subtracting less than one full surface to/from the sphere would result in a bump/dent on an otherwise spherical object.

 

In this computation, we compare the intermediate steps of the expansion and the contraction of the same original spherical volume, by adding and subtracting, respectively, the spherical surface to/from the spherical volume. Subsequently, we compare the final steps of the contraction and the expansion of the previously expanded and previously contracted spheres by subtracting and adding, respectively, the spherical surface from/to the spherical volume.

 

       DIMENSIONLESS SPHERICAL DIAMETER

 

           PRE-EXPANSION SPHERICAL RADIUS

 

             PRE-CONTRACTION SPHERICAL RADIUS

          PRE-EXPANSION SPHERICAL VOLUME

          PRE-EXPANSION SPHERICAL SURFACE

 

        POST-EXPANSION SPHERICAL VOLUME

 

         POST-EXPANSION SPHERICAL RADIUS

 

         POST-EXPANSION SPHERICAL SURFACE

 

        SUBSEQUENT CONTRACTION EXPANDED VOLUME

        SUBSEQUENT CONTRACTION EXPANDED RADIUS

 

       PRE-CONTRACTION SPHERICAL VOLUME

 

          PRE-CONTRACTION SPHERICAL SURFACE

 

         POST-CONTRACTION SPHERICAL VOLUME

 

        POST-CONTRACTION SPHERICAL RADIUS

 

        POST-CONTRACTION SPHERICAL SURFACE

 

       SUBSEQUENT EXPANSION CONTRACTED VOLUME

 

        SUBSEQUENT EXPANSION CONTRACTED RADIUS

 

    RATIO OF ORIGINAL RADII

   RATIO OF EXPANDED RADIUS TO CONTRACTED RADIUS

 

RATIO OF RADIUS OF SUBSEQUENTLY CONTRACTED RADIUS

      AFTER EXPANSION TO RADIUS OF SUBSEQUENTLY EXPANDED

RADIUS AFTER CONTRACTION

 

Comparison of the intermediate and the final volumes resulting from the above process shows that the two processes represent quite different paths. It can be said that, in this general example, neither the expansion/contraction process of a dimensionless sphere and the contraction/expansion process of a dimensionless sphere return the sphere to its original volume; and furthermore, the process paths are incongruent.

 

It is also noteworthy that a value greater than 3 (three) is required for positive volume to result in the contraction process. We can deduct from this observation, that while expansion is theoretically limitless, a finite nonzero boundary exists for contraction of a general dimensionless sphere resulting in positive volume. As the radius of the original spherical volume is increased, the incongruencies become less noticeable.

 

Consequently, we have demonstrated, with simple mathematics, the irreversibility of hysteresis which is observable in heat transfer, thermodynamics, electricity, magnetism, material strain, and other physical processes.

Now consider a sphere which expands by one surface to a volume and then contracts by one surface to a volume. It will be seen that the expansion/contraction process is irreversible. In this case, allow for elimination of units by setting the initial radius at the limit dimensionless value 3. Therefore, any dimensioned linear radius must be divided by 3 in order to find the associated linear scale factor. The transmission/reception pod associated with the radius will then be dimensionless e1/e in cross-sectional radius. The half wavelength for the transmission/reception pod is equal to the radius of its respective sphere.

 

      PRE-EXPANSION SPHERICAL RADIUS = 3

 

          INITIAL SPHERICAL VOLUME

                                  INITIAL SPHERICAL SURFACE

 

 

          INITIAL SPHERICAL VOLUME

 

            POST-EXPANSION SPHERICAL RADIUS

 

               EXPANSION WAVELENGTH

 

               INITIAL POD MAXIMUM SECTION RADIUS

 

          POST-EXPANSION POD MAXIMUM SECTION RADIUS

 

    POD SECTION STEP

 

         AREA OF POD SECTION STEP

 

             PRE-CONTRACTION SPHERICAL RADIUS

 

             PRE-CONTRACTION SPHERICAL VOLUME

 

                PRE-CONTRACTION SPHERICAL SURFACE

 

           POST-CONTRACTION SPHERICAL VOLUME

 

               POST-CONTRACTION SPHERICAL RADIUS

 

       POST-CONTRACTION SPHERICAL SURFACE

 

         CONTRACTION WAVELENGTH

 

              INITIAL POD MAXIMUM SECTION RADIUS

 

             POST-CONTRACTION POD MAXIMUM SECTION RADIUS

 

   POD SECTION STEP

 

                  AREA OF POD SECTION STEP

 

    RATIO OF EXPANDED TO CONTRACTED SECTION STEP AREA

 

         RATIO OF INITIAL TO EXPANDED/CONTRACTED SPHERICAL RADIUS

 

    RATIO OF INITIAL TO EXPANDED/CONTRACTED SPHERICAL SURFACE

 

    RATIO OF INITIAL TO EXPANDED/CONTRACTED SPHERICAL VOLUME

 

     RATIO OF EXPANSION TO CONTRACTION WAVELENGTH