Research On Fractional Derivative Based Constitutive Models Of Thermally Activated Amorphous Shape Memory Polymers

Shape memory polymers(SMPs)have become one kind of the most important and valuable new functional or smart materials in the last thirty years for their broad applications.The thermally activated SMPs have drawn considerable attention for their great variety,simple activated form,large recoverable deformation and rapid response.The establishment of an effective thermomechanical constitutive model lays a foundation for the broad and reliable applications of these materials.At present,the modelling research is still at the preliminary stage of exploration,which needs extensive and in-depth research.The main purpose of this study is to develop new constitutive models based on fractional derivative by using the viscoelastic approach.The main contents are as follows:Firstly,fractional models are used to describe the viscoelastic behaviors of thermally activated SMPs,which contain static behaviors such as relaxation modulus and creep compliance over a wide range of time and dynamic behaviors like storage modulus and loss factor over a wide range of frequency.At the same time,the fractional viscoelastic models are compared with the traditional integer-order viscoelastic models,Kohlrausch–Williams–Watts(KWW)model and Cole-Cole model.Secondly,the analytic partial derivatives of the Mittag-Leffler relaxation function which plays an important role in fractional models are complex to compute as well as the Mittag-Leffler relaxation function,thus a direct search method based on Powell's method is introduced to solve the minimization problem of nonlinear least-squares data fitting for the Mittag-Leffler function.A simple and effective method is provided for the determination of the initial values and an acceleration strategy is proposed for this direct search method.Furthermore,an effective algorithm is given for computing the timeconsuming Mittag-Leffler relaxation function and the effect of this algorithm on the numerical problems such as data fitting,numerical integration,numerical solutions of differential equations and integral equations are explored.Furthermore,a fractional vicoelastic model is utilized to predict the temperature-dependent free recovery behaviors of amorphous SMPs by introducing the time-temperature superposition principle(TTSP)and the change with the temperature of the time-temperature superposition shift factor is expressed by the Williams-Landel-Ferry(WLF)equations and Arrhenius law.Moreover,the effect of material parameter,heating rate and recovery temperature on the free recovery behaviors is studied.Finally,a multi-branch fractional thermoviscoelastic model is proposed to simulate and predict the temperature-dependent free recovery behaviors of amorphous SMPs such as acrylate-based network polymer,polyurethane and perfluorosulphonic acid ionomer(PFSA).The model is also valid for the triple shape memory effect and multi-shape memory effect.In addition,the model is proved effective for the simulation of the free recovery behaviors of SMPs composite by examining the simulations of the free recovery behaviors of a family of(meth)acrylate-based networks with different weight fractions of the crosslinking agent and the SMPs based syntactic foam consists of hollow glass microspheres dispersed in a shape memory polymer matrix.