On the appearance of fractional operators in non-linear stress–strain relation of metals

Abstract

Finding an accurate stress-strain relation, able to describe the mechanical behavior of metals during forming and machining processes, is an important challenge in several fields of mechanics, with significant repercussions in the technological field. Indeed, in order to predict the real mechanical behavior of materials, constitutive laws must be able to take into account elastic, viscous and plastic phenomena. Most constitutive models are based on empirical evidence and/or theoretical approaches, and provide a good prediction of the mechanical behavior of several materials.Here we present a non linear stress-strain relation based on fractional operators. The proposed constitutive law is based on integral formulation, and takes into account the viscoelastic behavior of the material and the inelastic phenomenon that appears when the stress reaches a particular yielding value. A specific case of the proposed constitutive law for imposed strain history is used to fit experimental data obtained from tensile tests on two kind of metal alloys. A best-fitting procedure demonstrates the accuracy of the proposed stress-strain relation and its results are compared to those obtained with some classical models. We conclude that the proposed model provides the best results in predicting the mechanical behavior for low and high values of stress/strain.

Advanced materials modelling via fractional calculus: challenges and perspectives

“…Ingman & Suzdalnitsky [43] have described the behaviour of a polymeric material using a fractional operator of time-dependent order, built as generalization of the Riemann-Liouville operator; in this case, the order varies with respect to the independent variable of the problem. Variableorder fractional operators have been used to capture nonlinear behaviour of metals and asphalt mixtures [44,45]. The advantages of using variable-order operators to tackle nonlinear problems are described by Patnaik et al [46], with focus on several application fields including nonlinear viscoelasticity.…”

An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

“…from which the poles of the system can be obtained. Resorting to the state space is a more elegant way to determine the poles of the system [8,20,28,29], but the presented process allows us to focus on the qualitative properties of the fractionally damped oscillator keeping the mathematical complexity to a minimum. From (6), one could think that a fractionally damped system has 2q poles but, in reality, only two of them-a complex conjugate pair [15], actually-are solutions to the original system (5), the rest are extraneous solutions due to the solving process.…”

“…Fractional models are specially advantageous for polymeric materials that show some level of dependence to frequency and arise naturally, for example, from certain motions of Newtonian fluids [3] or the molecular theories that predict the behaviour of certain type of polymeric materials [4]. In fact, fractional models are used to capture with more ease the viscoelastic nature of materials such as rubber or concrete [5] whose behaviour was previously modelled with a power law [6,7]; fractional operators appear in the non-linear stress-strain relation of metals [8]; and viscoelastic constitutive models based on fractional derivatives have been proposed to reproduce the time dependent behaviour of real materials [9][10][11][12][13][14].…”

On the nonlocal bending problem with fractional hereditariness

“…Among the various model of linear viscoelasticity, hereditary model based on the fractional calculus is able to represent the real time-dependent behavior of a wide variety of materials, like polymers [18], biological tissues [19], clays [20], non-newtonian fluids [5], rubbers [21], bones [22], and so on [23,24]. Moreover, these models are also recently used to model nonlinear time-dependent behavior [25][26][27]. For these capabilities viscoelastic models based upon fractional calculus will be used in this paper to obtain a versatile timedependent stress-strain relation useful to model several advanced bending problems where hereditary effects cannot be neglected.…”