Recent Ph.D. Theses:
Supervisor : Prof. R. O. Ogden
Modelling the mechanical behaviour of arterial soft tissues subject to volumetric growth.
Abstract: Studying the mechanics of soft biological tissue growth is of great importance, since a better understanding of the underlying physical laws that govern the mechanical behaviour of living tissues, such as arteries, will, in the future, have a considerable impact on the clinical techniques that are employed by doctors to fight the numerous diseases associated with the cardiovascular system. That is, the ultimate goal of researchers in biomechanics is to better understand the human body, so as to improve existing medical treatments for known diseases and to develop new ways of detecting these diseases.
This thesis seeks to review, firstly, some of the principal results concerning arterial wall mechanics that were acquired in the last few decades. This embodies knowledge from various fields, such as biology, medicine, physics and applied mathematics. Moreover, we tried to enlighten the reader on some of the important questions and difficulties encountered by researchers working on the development of theoretical models for the mechanical behaviour of arteries subject to growth.
Our main contribution to the field comes in the form of a new theoretical framework for the study of growth in this particular class of tissues. The primary goal of this new formulation is to overcome some of the problems arising from previously developed growth models. In a few words, the main advantage of the present growth model is not to require knowledge of an evolving stress-free configuration for the considered system, and to rely instead on a fixed residually-stressed reference configuration for the analysis of the mechanical response. This might, in fact, prove crucial for the advancement of biomechanics research since stress-free configurations of living soft tissues are not accessible in life and might not even exist in reality.
Supervisor : Prof. R. O. Ogden
Waves and surface stability in predeformed hyperelastic solids.
Abstract: Prestresses are a common occurrence in solids, as they can appear, for instance, during an assembly or curing process (elastomers), or during growth and remodelling (biological soft tissues), or during slow formation processes (rocks). One connected crucial question is: how large can a prestress or a prestrain be before the solid starts buckling? Seismic insulators, for instance, are typically constituted of a steel reinforced elastomer, which is able to support part of the weight of a building and to undergo very large deformations during seismic events; for obvious reasons, a static and dynamic analysis (including stability analysis) of these materials is required. The framework of incremental (infinitesimal) motions superimposed onto a large predeformation provides a natural theory to address these questions. The study of incremental motions – in particular, wave propagation – inside a predeformed material also leads to non-destructive evaluation analysis (the so-called acousto-
elastic effect). Clearly, large prestrains induce privileged directions (and thus
anisotropy) inside the deformed material, whose influence on the wave speed should not be neglected.
This Thesis proposes to study surface and interface waves, and the related topics of
surface and interface stability, in highly deformed materials. We are concerned with incompressible hyperelastic materials, typically used to model industrial elastomers,
natural rubbers, and soft biomaterials. We focus on problems in relation with
– the surface wave itself: here we study how the speed wave and the propagation direction are related to the prestress;
– the interface itself: here we compare the situation where a deformed solid is in
contact with a viscous fluid to the situation where it is in contact with vacuum, in terms of interface wave propagation;
– the material itself: here we develop a theory of incremental deformations for materials sensitive to magnetic fields and use it to study surface stability in the presence of a magnetic/mechanical coupling.
Supervisor : Dr. D.M. Haughton
The eversion and bifurcation of elastic cylinders
Abstract: In this Thesis we consider the eversion and bifurcation of both incompressible and compressible isotropic elastic cylinders.
To begin, we give a brief account of the basic equations of non-linear elasticity. We then study the basic eversion of hollow cylinders composed of both incompressible and compressible material for a variety of strain-energy functions and offer some analysis for the existence and uniqueness of the cylindrical everted state achieved.
Next, we study the effect of applying an incremental deformation to the basic everted state and formulate the bifurcation problem with the undeformed thickness ratio as a parameter. We study the bifurcation problem in detail for a variety of strain-energy functions and consider the effects of compressibility, initial tube thickness and mode numbers on the bifurcation produced. The bifurcation problems are solved numerically and we use the present problem to study two different numerical methods. We find that the standard determinantal method, extensively used in the past for elastic bifurcation problems, is not adequate for the problems considered in this Thesis, and thus adopt the Compound Matrix method. We compare both methods and give a derivation of the Compound Matrix method.
Supervisor : Prof. R.W. Ogden
Propagation, reflection and transmission of plane waves in pre-stressed elastic solids
Abstract: This Thesis is concerned with the effect of pure homogeneous strain, pre-stress and simple shear deformation on the propagation of homogeneous, surface and interfacial waves in elastic materials. Following a review of previous work regarding the propagation of infinitesimal plane waves in a half-space of incompressible material subject to pure homogeneous strain, the analysis is extended to the influence of pure homogeneous strain on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible isotropic elastic material. In general, the half-spaces consist of different material and are subject to different deformations.
For a certain class of constitutive laws it is shown that a homogeneous plane (SV) wave incident on the boundary from one half-space gives rise to a reflected wave (with angle of reflection equal to the angle of incidence) together with an interfacial wave in the same half-space, while in the other half-space two possibilities arise depending on the angle of incidence, the material properties and the magnitudes of the deformations in the two half-spaces. Either, (a) there is a transmitted (homogeneous plane) wave accompanied by an interfacial wave, or (b) there are two interfacial waves with equal speeds of propagation but different rates of (spatial) decay away from the boundary.
For a second class of constitutive laws similar behaviour is found for certain combinations of angle of incidence, material properties and deformations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and an interfacial wave, coupled with either possibility (a) or (b) in the second half-space. Equally, there may be two transmitted waves for each of the possible combinations of reflected and interfacial waves in the first half-space.
The effect of finite strain principal axis orientation on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material is then examined. Attention is focused on waves propagating in a principal plane of the deformation corresponding to simple shear.
For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence although its amplitude may vanish at certain discrete angles depending on the state of stress and deformation. Unlike the situation in which the underlying deformation is a pure homogeneous strain, however, the amplitude ratio of the deflected (plane harmonic) wave does not in general have unit magnitude, but is independent of the pre-stress. Moreover the angle of reflection differs from the angle of incidence.
For materials not in this special class, on the other hand, it is shown that two plane harmonic waves may be reflected when the angle of incidence lies within certain ranges of values (which depend on the shear deformation). Outside this range there is in general a single reflected wave, and a surface wave is generated.
This analysis is further used to study the effect of simple shear on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible elastic material, and, in particular, two half-spaces which form a twin in the sense that equal and opposite simple shears are applied to the two half-spaces.
For a special class of constitutive laws it is shown that an incident plane harmonic (shear) wave propagating in the plane of shear in one half-space gives rise to an interfacial wave in each half-space in addition to a reflected and a transmitted plane wave in the respective half-spaces for every angle of incidence, although the amplitudes of the waves may vanish at certain discrete angles (different for each type of wave) depending on the state of deformation.
For a specific material not in this special class, corresponding calculations for a particular value of shear show that the nature of the resulting waves is similar to that for the special class of materials. On the other hand, we note that for values of the shear beyond a critical value there are certain ranges of angles of incidence for which, instead, two homogeneous plane (shear) waves are reflected and also transmitted.
The dependence of the amplitudes of the reflected, transmitted and interfacial waves on the angle of incidence, the states of deformation and the material properties are illustrated graphically.
Supervisor : Dr. D.M. Haughton
Wrinkling problems for non-linear elastic membranes
Abstract: In this Thesis we study several examples of finite deformations of non-linear, elastic, isotropic membranes consisting of both incompressible and compressible materials which result in the membrane being wrinkled. To investigate the nature and occurrence of these wrinkled regions we adapt ordinary membrane theory by using a systematic approach developed by Pipkin (1986) and Steigmann (1990) which accounts for wrinkling automatically. In each problem considered, we employ the relaxed strain-energy function proposed by Pipkin (1986) and assume that the in-plane principal Cauchy stresses are non-negative.
A discussion of the basic equations for a membrane from the three-dimensional theory and the derivation of the relaxed strain-energy function from tension field theory are given. The resulting equations of equilibrium are then used to formulate various problems considered and solutions are obtained by analytical or numerical means for both the tense and wrinkled regions. In particular we consider the deformation of a membrane annulus of uniform thickness which is subjected to either a displacement or a stress on the inner and outer radii. We present the first analytical solution for such a problem, for incompressible and compressible materials, for both the tense and wrinkled regions. The first problem therefore provides a simple example illustrating the theory of Pipkin (1986).
The second problem studies an elastic, circular, cylindrical membrane which is inflated by an internal pressure and subjected to flexural deformation. The equations of equilibrium are solved numerically, two different solution methods being described, and the results are presented graphically showing the deformed cross-section of the cylinder for incompressible materials. Particular attention is given to the value of curvature at which wrinkling begins. An incremental deformation is also considered to investigate possible bifurcation solutions which could occur at some finite value of curvature.
The final problem considers two butt jointed, incompressible, elastic, circular cylindrical membranes of different material and geometric properties. In particular we fix the cylinders to have different initial radii which ensures that wrinkling will occur. The composite cylinder is inflated and subjected to axial loading on either end. This deformation may have useful applications in surgery as it could be considered as a first approximation model for arterial grafts, the wrinkled surface having important
implications in the formation of blood clots as blood flows through such a region. Again, the equations of equilibrium are solved numerically and graphical results of the deformed, axial length against the deformed radius for a range of values of the parameters are given showing the tense and wrinkled regions.
Supervisor : Prof. R.W.Ogden
On deformations of compressible hyperelastic material
Abstract: We consider the character of several finite deformations of compressible, isotropic, nonlinear hyperelastic materials, specifically azimuthal shear of a thick-walled circular cylindrical tube, the bending deformation of a rectangular block, and axial shear of a thick-walled circular cylindrical tube. For each problem the equilibrium equations are applied to the special case of isochoric deformation, and explicit necessary and sufficient conditions on the strain energy function for the material to admit such deformation are obtained. These conditions are examined for several strain-energy functions and in each case complete solutions of the equilibrium equations re obtained. The predictions of the shear response for different strain-energy functions are compared using numerical results to show the dependence of the applied shear stress on the resulting macroscopic deformation. It is then shown how consideration of isochoric deformations in compressible elastic materials provides a means of generating classes of strain-energy functions for which closed-form solutions can be found for incompressible materials. For the problem of bending deformation we find that isochoric deformation is not possible in a compressible material. The conditions for non-isochoric bending deformation to be admitted by the equilibrium equations are then examined for each of three classes of compressible isotropic materials. Explicit solutions for each case are then derived. Finally, we consider an incremental displacement superimposed on the azimuthal shear of a circular cylindrical tube. Numerical results are obtained to show the incremental displacement and nominal stresses for a special material when the internal boundary is subject to an incremental displacement.
Supervisor : Prof. R.W. Ogden
Bifurcation and vibration of a surface-coated elastic block under flexure
Abstract: The behaviour of a surface-coated rectangular, non-linearly elastic block subject to (plane-strain) flexure is investigated in this Thesis. We consider a rectangular block of incompressible isotropic elastic material coated with a thin elastic film on part of its boundary. Initially, the bulk material undergoes non-homogeneous deformation and the equilibrium of the coated body is examined on the basis of the elastic surface coating theory derived by Steigmann and Ogden (1997a). Incremental displacements are then superimposed on the finitely deformed configuration in order to study possible bifurcation of the deformed block. Numerical bifurcation results pertaining to two particular strain-energy functions (for the bulk material) and a general energy function (for the coating material) are subsequently obtained. These results allow the influence of the surface coating on the bifurcation behaviour of the block to be determined and assessed with reference to corresponding results for an uncoated block. Next, use is made of the dynamic equivalent of the static surface coating theory developed by Ogden and Steigmann (1999), to establish incremental equations of motion for the coated block. Corresponding incremental governing equations for an uncoated, pre-flexed block then emerge as a special case. The resulting frequency equations are solved numerically, again on specialisation of the form of strain-energy function. The numerical vibration results then provide evidence of the effect of surface coating on the dynamic behaviour of the considered coated block relative to the uncoated case. Finally, we turn our attention to the (non-linear) shear response of bonded elastic bodies. We examine the plane-strain problem of a rectangular compressible isotropic elastic block bonded to two rigid parallel plates. The deformation behaviour of the block is described by applying minimum energy and maximum complementary energy principles to obtain upper and lower bounds on the shear stress-strain relationship. Although maximum and minimum principles are not generally justifiable in non-linear elasticity we show that under certain conditions they are applicable and, for a particular form of strain-energy function, derive explicit energy bounds which we illustrate graphically.