A single integral approach to fractional order non-linear hereditariness

Abstract

In this study a non-linear approach to the hereditariness of complex materials and soft matter is faced within the context of fractional differential calculus. The non-linear dependence of the creep (relaxation) function on the level of stress (strain) observed in experimental tests conducted at the Bio/NanoMechanics for Medical Sciences Lab of Palermo University on ligaments and tendons of the human knee has been discussed with a multiplicative decomposition of the material characteristic function. This approach, followed by a proper non-linear transform of the material state variables yields a Volterra convolution in terms of the transformed variables and a non-linear model involving fractional-order operators with specified relations among the integro-differentiation orders has been obtained. Some numerical applications showing the methodology for some test functions have been reported in the text.