# New Mathematical Model for Correlation Between Tensile Elastic Modulus and Shore “A” and “00” Hardness for Flexible Polymers

## TL;DR

This paper introduces a new mathematical model that links the tensile elastic modulus of flexible polymers to their Shore hardness measurements, enabling quick and practical estimation for industrial and medical applications.

## Contribution

A novel piecewise exponential correlation model is developed to estimate tensile elastic modulus from Shore hardness measurements with minimized error.

## Key findings

- The model achieves a maximum relative error of 13.4% for the OO scale and 7% for the A scale.
- The model is monotonic and provides a practical method for estimating elastic modulus from non-destructive hardness measurements.
- The model is particularly useful for industrial applications and dental surgery simulations.

## Abstract

The paper presents the development of a correlation model for initial tensile elastic modulus for flexible polymers as a function of Shore hardness in OO and A scale based on measurement. Measured polymers are in groups of silicone rubber, nitrile butadiene rubber (NBR), thermoplastic polyurethane (TPU) and silicone. The model is composed of piecewise exponential functions with fixed coefficients chosen to minimize the S2 error norm and absolute value of relative error at the measured data points. Every chosen section of the hardness scale has one exponential function correlating the hardness to tensile elastic modulus with the argument in the form of a polynomial up to the fourth degree. The coefficients for the polynomial arguments were determined by enforcing interpolation conditions in a chosen set of points in the logarithmic scale for the elastic modulus. The correlation model possesses C0 continuity. For each material, five specimens were used for hardness measurements and five for the elastic modulus testing. The correlation model gives a positive value for elastic modulus of 0 for hardness, and a “finite”, “reasonable” value of 100 for hardness and is monotonic. Tensile properties were evaluated using true stress and logarithmic (Hencky) strain, with iterative correction of the changing cross-sectional area to account for large strain. The maximum relative error achieved in the correlation model for the OO scale is 13.4%, while for the A scale it is 7%. The developed model provides a practical and rapid method for estimating the initial tensile elastic modulus from non-destructive hardness measurements and is particularly useful in industrial applications and in the development of material models for dental surgery simulations.