Some dynamic mechanical properties of apple fruits and their use in the solution of an impacting contact problem of spherical fruit

Complex compression moduli were determined for five groups of apples by subjecting cylindrical specimens to sinusoidal forcing at cyclic rates of 50 to 365 cycles per second. The five groups of apples consisted of Red Delicious, Winesap, and Golden Delicious at harvest and Winesap and Golden Delicious after about four months’ storage. Ten tests at each of seven frequencies were conducted for each group of apples.

Statistical analyses of the complex compression moduli showed the stress-strain relations for the various groups of apples to be frequency dependent. Due to scatter, Maxwell model approximations were the best mechanical model representations that could be developed directly. These, however, appeared to be sufficiently good approximations for the data. A statistical curve fitting procedure resulted in expressions for relaxation moduli which were equivalent to Maxwell model representations with element constants slightly larger than those obtained by the direct method. Several tables and curves are given to summarize and clarify these experimental results.

The two groups of apples taken from storage were also tested in sinusoidal shear but shear moduli obtained were believed to be low by an unknown percentage due to the fact that a pure shear state could not be very closely approximated and it was not possible to insure that there was no specimen slip in the test fixture. Shear moduli were, however, roughly the magnitude expected based on the knowledge of the compression moduli and the imperfect shear state.

Poisson’s ratio values for the three groups of apples tested at harvest condition were determined by Chappel* and used in the contact problem portion of this thesis.

Contact problems involving one apple falling onto another supported underneath and an apple falling onto a rigid surface were solved for the approach of the bodies, surface indentations, surface pressures, and internal stresses from the time of initial contact to the time of maximum indentation. A theoretical analysis was made and a computer program written which computed these quantities with the input being the element constants of the Maxwell material, Poisson’s ratio, the radius of the apple at the contact point, the apple weight, and the drop height. Computer time using an IBM 7040 digital computer was short (about four minutes) except for computation of internal stresses which took several hours depending on the number of points at which stress was being computed. Comparison of computer results and published elastic solutions validated the method. Example problems worked to illustrate the method revealed that high internal stresses (about 200 psi) are caused by a two-inch drop of a representative apple. Higher drop heights produce a somewhat higher maximum stress and a much larger high stress region. A number of curves are presented to illustrate the approach of the impacting bodies, contact stress, and internal stresses for the example problems.

The experimental equipment and techniques described and the solution to the impact problem described are not limited to apple flesh or biological materials and possibly will be even more useful for other materials.