Measurement and evaluation of the apparent modulus of elasticity of apple based on Hooke’s, Hertz’s and Boussinesq’s theories

• Apple bruising is an important issue.
• Three methods were used to measure modulus of elasticity.
• Apple handling is considered.
• Hooke’s, Hertz’s and Boussinesq’s theories were addressed.

Awareness on the mechanical properties of agricultural products is necessary in order to estimate and predict the deformation of viscoelastic materials under external loads for system design, transport, processing and packaging. The aim of this study was to evaluate three theories (i.e. Hooke, Hertz and Boussinesq) on the apparent modulus of elasticity of some apple varieties i.e. Golden Delicious, Red Delicious and Granny Smith. The theoretical results were analyzed in a factorial experiment with completely randomized design with 9 treatments and 15 replicates. The results showed the practical usability of Hertz’s theory with better prediction accuracy whilst the Boussinesq’s theory showed a significant difference of predicted modulus of elasticity compared to other theories and the values reported in some publications. Although, the Hook’s theory enables the identification of a bio-yield point on its force–deformation curve, but Hertz’s theory was recommended as the most appropriate method due to easiness of working on a cylindrical sample of apple and the closer agreement with reality. Based on the results of this study, Golden Delicious and Granny Smith varieties had the lowest and the highest modulus of elasticity as 2.211 MPa and 3.431 MPa, respectively. This difference shows the firmness of apple varieties and different tissue responses to external loads and forces accordingly.

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