This monograph provides a comprehensive overview of the author's work on the fields of fractional calculus and waves in linear viscoelastic media, which includes his pioneering contributions on the ...applications of special functions of the Mittag-Leffler and Wright types.It is intended to serve as a general introduction to the above-mentioned areas of mathematical modeling. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to delve further into the subject and explore the research literature given in the huge general bibliography.This book is likely to be of interest to applied scientists and engineers.
In the field of Soft Robotics, viscoelasticity has been proved beneficial for human assistance applications. The human skeletal muscle system, as well as many soft materials commonly used in soft ...robotic applications, have viscoelastic properties. Viscoelasticity can be modelled using a set of equations known as the Linear Viscoelastic Models (LVMs). This modelling approach has two main limitations: high mathematical complexity and high computational cost. Here, these limitations are addressed in two ways. Firstly, the Piecewise Linearisation method is used to reduce the mathematical complexity of LVMs. Secondly, a modelling approach based on feedforward artificial neural networks (ANNs) is used to reduce the computational cost. The aim of both modelling approaches is to describe the non-linear, strain-dependent, and time-dependent stress response of seven thermoplastic elastomers. On the one hand, the implementation of the Piecewise Linearisation method yielded the PL-SLS model and the PL-Wiechert model. Both models were successful in predicting the viscoelastic behaviour of the materials, outperforming similar modelling tools documented in the literature. On the other hand, four different architectures of ANN models are developed, categorized in rate-dependent and rate-independent. Results highlight the rate-dependent architecture as the most suitable. The ANN models achieved a similar prediction performance as the PL models. The ANN model for the natural rubber material is further validated in a real-time simulation environment, in Simulink. This soft material is found to be the best candidate to imitate the mechanical properties of the human tendon. On the one hand, the performance prediction of the ANN models is adequate for a sine wave strain input, when the strain rate is constant. On the other hand, the response of the ANN model is unstable under variable strain rates. This highlights an important limitation of the training set used for developing the ANN models, which only contains data for three different strain rates. Finally, the three modelling tools developed in this research are a direct improvement to current modelling approaches. Nonetheless, a richer training set is required to improve the ANN models real-time response.
This text is a guide how to solve problems in which viscoelasticity is present using existing commercial computational codes. The book gives information on codes structure and use, data preparation ...and output interpretation and verification. The first part of the book introduces the reader to the subject, and to provide the models, equations and notation to be used in the computational applications. The second part shows the most important Computational techniques: Finite elements formulation, Boundary elements formulation, and presents the solutions of Viscoelastic problems with Abaqus.
Rubberlike materials exhibit strong rate-dependent mechanical response which manifests itself in creep and relaxation tests as well as in the hysteresis curves under cyclic loading. Unlike linear ...viscoelasticity, creep and relaxation response of rubber is nonlinear and amplitude-dependent. Within this context, the contribution of this work is three fold. (i) On the experimental side, the characterization of equilibrium and non-equilibrium responses are carried out by means of uniaxial and equibiaxial extension tests. Also performed are the creep and relaxation experiments at various stress and strain levels. (ii) On the theoretical side, we extend the well-known eight-chain model via incorporating a simple yet instrumental tube-constraint term composed of the second invariant into the non-affine network contribution reflecting the ground state equilibrium response. For the non-equilibrium response, we propose a new evolution equation for the creep/relaxation behavior of rubberlike materials based on a relaxation kinetics of a single polymer chain. The geometric non-linearity is incorporated via the finite deformation kinematics based on the multiplicative split of the deformation gradient into elastic and viscous parts, whereas the volumetric and isochoric split of the deformation gradient is entirely discarded. The rheology uses a nonlinear spring responsible for equilibrium elastic response in parallel to n number of Maxwell elements, leading to a generalized Maxwell-Wiechert viscoelastic solid. (iii) The algorithmic implementation of the model features the spectral decomposition of the respective terms and is demonstrated within the context of the finite element method. The developed model is validated by fitting both the elastic and viscoelastic model responses with respect to the experimental data in the sense of uniaxial, (equi)biaxial extensions, and pure shear tests. Relaxation and creep behavior of the model are thoroughly assessed. Also presented in the manuscript is the capability and the performance of the model in the face of a relevant non-homogeneous boundary value problem. The quality of the findings earns the model vast utilization areas from an engineering perspective.
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