arXiv subjects

Tom Shearer

Publications and source records attributed to Tom Shearer.

15 recordsLinked to original sources

arXiv2025

A microstructural model of transversely isotropic, fibre-reinforced hydrogels

Fibre-reinforced hydrogels are promising materials for biomedical applications due to their strength, toughness, and tunability. However, it remains unclear how to design fibre-reinforced hydrogels for use in specific applications due to the lack of a flexible modelling framework that can predict and hence optimise their behaviour. In this paper, we present a microstructural model for transversely isotropic fibre-reinforced hydrogels that captures the specific geometry of the fibre network. The model also accounts for slack in the initial fibre network that is gradually removed upon deformation. The mechanical model for the fibre network is coupled to a nonlinear poroelastic model for the hydrogel matrix that accounts for osmotic stress. By comparing the model predictions to data from unconfined compression experiments, we show that the model can capture J-shaped stress-strain curves and time-dependent creep responses. We showcase how the model can be used to guide the design of materials for artificial cartilage by exploring how to maximise interstitial fluid pressure. We find that fluid pressurisation can be increased by using stiffer fibres, removing slack from the fibre network, and reducing the Young's modulus of the hydrogel matrix. Finally, a high-level and open-source Python package has been developed for simulating unconfined compression experiments using the model.

cond-mat.soft↗

arXiv2024

Exploring natural variation in tendon constitutive parameters via Bayesian data selection and mixed effects models

Combining microstructural mechanical models with experimental data enhances our understanding of the mechanics of soft tissue, such as tendons. In previous work, a Bayesian framework was used to infer constitutive parameters from uniaxial stress-strain experiments on horse tendons, specifically the superficial digital flexor tendon (SDFT) and common digital extensor tendon (CDET), on a per-experiment basis. Here, we extend this analysis to investigate the natural variation of these parameters across a population of horses. Using a Bayesian mixed effects model, we infer population distributions of these parameters. Given that the chosen hyperelastic model does not account for tendon damage, careful data selection is necessary. Avoiding ad hoc methods, we introduce a hierarchical Bayesian data selection method. This two-stage approach selects data per experiment, and integrates data weightings into the Bayesian mixed effects model. Our results indicate that the CDET is stiffer than the SDFT, likely due to a higher collagen volume fraction. The modes of the parameter distributions yield estimates of the product of the collagen volume fraction and Young's modulus as 811.5 MPa for the SDFT and 1430.2 MPa for the CDET. This suggests that positional tendons have stiffer collagen fibrils and/or higher collagen volume density than energy-storing tendons.

stat.AP↗

arXiv2024

The effects of fibre spatial distribution and relative orientation on the percolation and mechanics of stochastic fibre networks: A model of peptide hydrogels

The structures of fibre networks can vary greatly due to fibre interactions during formation. We have modified the steps of generating Mikado networks to create two new model classes by altering the spatial distribution and relative orientation of their fibres to mimic the structures of self-assembling peptide hydrogels (SAPHs), whose physical properties depend strongly on their fibres' interactions. The results of our models and experiments on a set of beta-sheet forming SAPHs show that modifying a network's structure affects the percolation threshold and the mechanical behaviour of the material, both near percolation and at higher densities.

cond-mat.soft↗

arXiv2024

The elastic stored energy of initially strained, or stressed, materials: restrictions and third-order expansions

A large variety of materials, widely encountered both in engineering applications and in the biological realm, are characterised by a non-vanishing internal stress distribution, even in the absence of external deformations or applied forces. These initial stresses are due to initial strains or microstructural changes, such as thermal expansion, manufacturing processes, volumetric growth, remodelling, etc. A common constitutive choice for modelling such materials extends the classical approach in the field theory of continuum mechanics to include, explicitly, the initial stress or strain in the elastic stored energy function. Here, we discuss why these energy functions need to satisfy some restrictions to avoid unphysical behaviours, such as non-conservation of energy, {and we derive the required restrictions from the classical assumptions of elasticity.} To illustrate their need, we perform a rigorous asymptotic expansion for proving that these restrictions on stored energy functions that depend on the strain and initial stress are required for consistency with strain energy functions of classical third-order weakly nonlinear elasticity.

cond-mat.soft↗

arXiv2023

Tensor decomposition for modified quasi-linear viscoelastic models: towards a fully nonlinear theory

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic Modified Quasi-Linear Viscoelastic models. We show how to formulate the constitutive equation by using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the transversely isotropic case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the Modified Quasi-Linear Viscoelastic framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original Quasi-Linear Viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the Modified Quasi-Linear Viscoelastic model, with two examples (uni-axial extension and perpendicular shear).

cond-mat.soft↗

arXiv2022

Bayesian inference on a microstructural, hyperelastic model of tendon deformation

Microstructural models of soft tissue deformation are important in applications including artificial tissue design and surgical planning. The basis of these models, and their advantage over their phenomenological counterparts, is that they incorporate parameters that are directly linked to the tissue's microscale structure and constitutive behaviour and can therefore be used to predict the effects of structural changes to the tissue. Although studies have attempted to determine such parameters using diverse, state-of-the-art, experimental techniques, values ranging over several orders of magnitude have been reported, leading to uncertainty in the true parameter values and creating a need for models that can handle such uncertainty. We derive a microstructural, hyperelastic model for transversely isotropic soft tissues and use it to model the mechanical behaviour of tendons. To account for parameter uncertainty, we employ a Bayesian approach and apply an adaptive Markov chain Monte Carlo algorithm to determine posterior probability distributions for the model parameters. The obtained posterior distributions are consistent with parameter measurements previously reported and enable us to quantify the uncertainty in their values for each tendon sample that was modelled. This approach could serve as a prototype for quantifying parameter uncertainty in other soft tissues.

physics.bio-ph↗

arXiv2021

Silence is bliss in a Platonic relationship

We describe an effective active cloaking strategy for the scalar Helmholtz equation in three dimensions where multipole active sources are located at the vertices of the Platonic solids. A "silent zone" is created interior to the imaginary Platonic solid and only the incident field remains in a defined region exterior to the silent zone and active source configuration. This distribution of the sources ensures that the implementation of the cloaking strategy is extremely efficient. In particular, once the multipole source amplitudes required at a single source location are determined, the other source amplitudes can be calculated by simple post-processing involving multiplication of the multipole source vector by a rotation matrix. The general nature of the problem means that the technique is relevant to any scalar wave field, including both acoustics and electromagnetism.

physics.class-ph↗

arXiv2021

A microstructural model of tendon failure

Collagen fibrils are the most important structural component of tendons. Their crimped structure and parallel arrangement within the tendon lead to a distinctive non-linear stress-strain curve when a tendon is stretched. Microstructural models can be used to relate microscale collagen fibril mechanics to macroscale tendon mechanics, allowing us to identify the mechanisms behind each feature present in the stress-strain curve. Most models in the literature focus on the elastic behaviour of the tendon, and there are few which model beyond the elastic limit without introducing phenomenological parameters. We develop a model, built upon a collagen recruitment approach, that only contains microstructural parameters. We split the stress in the fibrils into elastic and plastic parts, and assume that the fibril yield stretch and rupture stretch are each described by a distribution function, rather than being single-valued. By changing the shapes of the distributions and their regions of overlap, we can produce macroscale tendon stress-strain curves that generate the full range of features observed experimentally, including those that could not be explained using existing models. These features include second linear regions occurring after the tendon has yielded, and step-like failure behaviour present after the stress has peaked. When we compare with an existing model, we find that our model reduces the average root mean squared error from 4.15MPa to 1.61MPa, and the resulting parameter values are closer to those found experimentally. Since our model contains only parameters that have a direct physical interpretation, it can be used to predict how processes such as ageing, disease, and injury affect the mechanical behaviour of tendons, provided we can quantify the effects of these processes on the microstructure.

q-bio.TO↗

arXiv2018

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in transversely isotropic, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

cond-mat.soft↗

arXiv2016

A new restriction for initially stressed elastic solids

We introduce a fundamental restriction on the strain energy function and stress tensor for initially stressed elastic solids. The restriction applies to strain energy functions $W$ that are explicit functions of the elastic deformation gradient $\mathbf{F}$ and initial stress $\boldsymbol τ$, i.e. $W:= W(\mathbf F, \boldsymbol τ)$. The restriction is a consequence of energy conservation and ensures that the predicted stress and strain energy do not depend upon an arbitrary choice of reference configuration. We call this restriction: initial stress reference independence (ISRI). It transpires that almost all strain energy functions found in the literature do not satisfy ISRI, and may therefore lead to unphysical behaviour, which we illustrate via a simple example. To remedy this shortcoming we derive three strain energy functions that do satisfy the restriction. We also show that using initial strain (often from a virtual configuration) to model initial stress leads to strain energy functions that automatically satisfy ISRI. Finally, we reach the following important result: ISRI reduces the number of unknowns of the linear stress tensor of initially stressed solids. This new way of reducing the linear stress may open new pathways for the non-destructive determination of initial stresses via ultrasonic experiments, among others.

cond-mat.soft↗

arXiv2016

Maths in Medicine: How to Survive a Science Fair

When talking to secondary school students, first impressions are crucial. Accidentally say something that sounds boring and you'll lose them in seconds. A physical demonstration can be an eye-catching way to begin an activity or spark off a conversation about mathematics. This is especially true in the context of an event like a science fair where there are hundreds of other exhibitors and stands, possibly involving loud music and/or dancing robots! In this article we describe three devices that were built to illustrate specific physical phenomena that occur in the human body. Each device corresponds to a simple mathematical model which contains both elements that are accessible to pupils in the early years of secondary education and more challenging mathematical concepts that might appeal to A-level students. Two of the devices relate to the Windkessel effect, a physical phenomenon that regulates blood flow, and the third demonstrates the elastic properties of ligaments and tendons.

physics.ed-ph↗

arXiv2015

Modelling afferent nerve responses to bladder filling

A sensation of fullness in the bladder is a regular experience, yet the mechanisms that act to generate this sensation remain poorly understood. This is an important issue because of the clinical problems that can result when this system does not function properly. The aim of the study group activity was to develop mathematical models that describe the mechanics of bladder filling, and how stretch modulates the firing rate of afferent nerves. Several models were developed, which were qualitatively consistent with experimental data obtained from a mouse model.

q-bio.TO↗

arXiv2015

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on the geometrical arrangement of their fibrils is derived. The distribution of the crimp angles of the fibrils is used to determine the stress-strain response of a single fascicle, and this stress-strain response is used to determine the form of the strain energy function, the parameters of which can all potentially be directly measured via experiments - unlike those of commonly used strain energy functions such as the Holzapfel-Gasser-Ogden (HGO) model, whose parameters are phenomenological. We compare the new model with the HGO model and show that the new model gives a better match to existing stress-strain data for human patellar tendon than the HGO model, with the average relative error in matching this data when using the new model being 0.053 (compared with 0.57 when using the HGO model), and the average absolute error when using the new model being 0.12 MPa (compared with 0.31 MPa when using the HGO model).

q-bio.TO↗

arXiv2015

A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrils

A new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout its radius is calculated and used to determine the form of the strain energy function. The new constitutive law is used to model uniaxial extension test data for human patellar tendon and is shown to provide an excellent fit, with the average relative error being 9.8%. It is then used to model shear and predicts that the stresses required to shear a tendon are much smaller than those required to uniaxially stretch it to the same strain level. Finally, the strain energy function is used to model ligaments and tendons whose fascicles are helical, and the relative effects of the fibril helix angle, the fascicle helix angle and the fibril crimp variable are compared. It is shown that they all have a significant effect; the fibril crimp variable governs the non-linearity of the stress-strain curve, whereas the helix angles primarily affect its stiffness. Smaller values of the helix angles lead to stiffer tendons; therefore, the model predicts that one would expect to see fewer helical sub-structures in stiff positional tendons, and more in those that are required to be more flexible.

q-bio.TO↗

arXiv2012

Employing pre-stress to generate finite cloaks for antiplane elastic waves

It is shown that nonlinear elastic pre-stress of neo-Hookean hyperelastic materials can be used as a mechanism to generate finite cloaks and thus render objects near-invisible to incoming antiplane elastic waves. This approach appears to negate the requirement for special cloaking metamaterials with inhomogeneous and anisotropic material properties in this case. These properties are induced naturally by virtue of the pre-stress. This appears to provide a mechanism for broadband cloaking since dispersive effects due to metamaterial microstructure will not arise.

physics.class-ph↗