# Mathematical models of inhomogeneous nonlinear viscoelastic solids and associated applications.

Lead Research Organisation: University of Manchester
Department Name: Mathematics

### Abstract

Constitutive models are essential in order to accurately model the way that materials deform under applied load. In terms of mechanics they typically relate stress (force per unit area) to strain or rate of strain. "Linear" materials are governed by simple linear relations: stress is proportional to strain (Hookean solids) or stress is proportional to rate of strain (Newtonian fluids). Linear viscoelastic materials behave somewhere between these two idealized media. These models typically apply to materials that undergo small deformations or small rates of strain. However they do not adequately describe a broad range of materials that include rubber and other elastomers of interest, foams and soft tissues predominantly because they can undergo large deformation but also because their material reponse is not linear. This behavior is even more difficult to capture in models when the medium in question is inhomogeneous, e.g. it may be filled with inclusions or nano-fillers. Reasons to include such fillers are usually associated with a desire to improve certain types of material response (stiffness, conductivity, etc.)

This project will focus on developing the fundamental mathematical theory of nonlinear viscoelasticity associated with viscoelastic solids, and particularly those that are inhomogeneous. Since the 1960s a small number of theories have been proposed that model specific nonlinear viscoelastic materials under certain types of deformation. These vary in complexity, ranging from simple mathematical expressions to complex formulations. A first objective will therefore involve determining the links between these models: where they overlap and where they do not and how they may converge to each other in certain limits. These linkages are currently not understood and a clear understanding would be extremely beneficial to communities in applied mathematics and materials science.

The student will also investigate how certain types of nonlinear viscoelastic behavior (e.g. strain dependent relaxation) can be accommodated by certain models but not others. In particular filled (inhomogeneous) elastomers often exhibit strong nonlinear behavior that is not present in homogeneous materials. In order to accommodate this strongly nonlinear behavior, new constitutive models will be developed. The starting point of the models will be quasi-linear viscoelasticity, which is viewed as a useful starting point in terms of a balance between being able to accurately represent certain types of behaviour and also being able to be implemented in computational models. Chiefly, the notion of strain-dependent relaxation will be investigated in these models and their ability to fit experimental data on a range of deformation modes of nonlinear materials. Of specific interest is existing experimental data associated with Syntactic foams, materials that are widely used in an array of applications ranging from aerospace, marine, sportswear and non-destructive evaluation. The constitutive models will be incorporated into open-source finite element software.

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### Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R513131/1 01/10/2018 30/09/2023
2291505 Studentship EP/R513131/1 01/10/2019 01/09/2020 Joshua Gillis