Hooke’s law and elastic constants are introduced. The symmetry of the elastic constant tensor follows from the symmetry of stress and strain tensors and the elastic energy density. The maximum number of independent elastic constants is 21 before crystal symmetry is considered, and this leads to the introduction of matrix notation. Neumann’s principle reduces the number of independent elastic constants in different crystal systems. It is proved that in isotropic elasticity there are only two independent elastic constants. The directional dependences of the 3 independent elastic constants in cubic crystals are derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3 involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in noncubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensor, and compatibility stresses.

Mechanical properties of crystalline materials are almost always dominated by the defects within them. The ability to shape metals into pipes, girders and furniture stems from the generation, motion and interaction of these defects. Defects are also the agents of chemical changes within crystals, enabling mass transport by atomic diffusion and changes of phase. Defects distort the crystal and these distortions enable defects to interact over large distances. The theory of elasticity is used to describe these interactions. Assuming no familiarity with the theory, this book introduces the reader to linear elasticity and its application to point defects, dislocations and cracks. A unique feature of the book is the attention given to the atomic structure of defects and its influence on their properties and their elastic fields. Where it is available brief biographical information is provided about prominent contributors to the field. This textbook is written for postgraduate students in physics, engineering and materials science. It is very likely that even those students with some knowledge of elasticity and defects will find much that is new to them in this book.There are exercises to help the student check their understanding as they work through each chapter. The student is guided through more advanced problems at the end of each chapter. Worked solutions to all exercises and problems are available to course instructors from the OUP website. The last chapter describes four technologically important areas requiring fundamental research, with suggestions for possible PhD projects.

The Cohesion Theory of the ascent of water in trees is a quiet triumph of modern science. Besides hydrodynamics, the physics of transpiration involves capillarity, evaporation and osmosis - phenomena which all have a history of considerable theoretical confusion. The aim of this paper is to supplement existing accounts of this physics in the plant science literature.

There are widespread misconceptions about the metallic bond and properties of metals in text-books and online resources in chemical education. This is surprising in view of the evident importance of metals and alloys in the modern World. Contrary to what is widely stated in the chemistry literature, Encyclopedia Britannica online, Wikipedia, ChatGPT, Google Bard and a recently published infographic on the Royal Society of Chemistry Web site, the electrostatic attraction between ion cores and free electrons in a simple metal is not sufficient to prevent it from disintegrating into its constituent atoms. It is also not true that localization of electrons is the reason for the difference between electronic conduction in metals and most insulators. The electrical resistance of metallic crystals is due to deviations from perfect structural and chemical order. The treatment in the same sources of processes within a crystalline metal when its shape is changed permanently bear no relation to experimental reality. Using almost no mathematics this article offers an insight at an accessible level into current thinking in chemistry and physics about metallic bonding and the properties of metals for teachers and students of chemistry in high schools and universities.

This chapter is based on Eshelby’s static energy-momentum tensor which results in an integral expression for the configurational force on a defect. After elucidating the concepts of a configurational force and an elastic singularity the mechanical pressure on an interface, such as a twin boundary or a martensitic interface, is derived. Eshelby’s force on a defect is derived using both physical arguments and more formally using classical field theory. It is equivalent to the J-integral in fracture mechanics. The Peach-Koehler force on a dislocation is rederived using the static energy-momentum tensor. An expression for an image force is derived, where a defect interacts with a free surface.

In the 1960s, it was discovered in the former Soviet Union that short pulses of intense electric currents through metallic wires brought about temporary drops in the flow stress of the metal as it is deformed plastically. This became known as the electroplastic effect, or electroplasticity. After more than 50 years of experimental and theoretical research, no consensus has emerged as to the mechanism of the effect. Following a brief review of the principal experimental results, we show that when a current flows through a metal the ionic cores of atoms of the metal experience a force equal to the Lorentz force on the conduction electrons, which arises from the magnetic field created by the current. This is the origin of the pinch effect in metals. We then present a new theory of electroplasticity based on mechanical stresses created by pulsing the current as a result of electromagnetic induction. Unlike earlier theories, the rate of change of the current is treated explicitly in a dynamic version of the pinch effect. Pulses of normal and shear stresses arise with a magnitude that depends on the rate at which the current changes during a pulse and a small number of other well defined variables. Unlike earlier theories which focused on the maximum of the current during a pulse, this new theory highlights the time dependence of the current pulse as well as its maximum value. Experiments to test the theory are suggested. Four mechanisms proposed earlier for the electroplastic effect are reviewed critically in the appendices. They are dislocation unpinning in a magnetic field, electromigration of dislocations, Joule heating, and the static pinch effect. We show that the physics of the first two mechanisms is unsound and the second two cannot explain most experimental observations.

High speed dislocations have long been identified as the dominant feature governing the plastic response of crystalline materials subjected to high strain rates, controlling deformation and failure in industrial processes such as machining, laser shock peening, punching, drilling, crashworthiness, foreign object damage, etc. Despite decades of study, the role high speed dislocations have on the materials response remains elusive. This article reviews both experimental and theoretical efforts made to address this issue in a systematic way. The lack of experimental evidence and direct observation of high speed dislocations means that most work on the matter is rooted on theory and simulations. This article offers a critical review of the competing theoretical accounts of high speed mechanisms, their underlying hypothesis, insights, and shortcomings, with particular focus on elastic continuum and atomistic levels. The article closes with an overview of the current state of the art and suggestions for key developments in future research.

There is clear evidence in the literature that iron segregates to the interface of second phase particles (SPPs) in unirradiated Zr-Nb alloys, and that it does not do so in the presence of radiation damage. In this work, a discrete dislocation plasticity model is developed that takes into account the long-range stress field of the SPP interface. A simple analytical model is also outlined, providing an upper bound for estimating the amount of interstitial segregation. The model provides a possible mechanism to explain both the iron segregation to coherent SPPs and its subsequent loss after irradiation. Qualitatively, the model proved to be insensitive to variations of all geometrical and computational parameters, allowing for general conclusions to be drawn. The model suggests that the segregation originates from a tensile field of order 1 GPa induced by the dislocations generated during the plastic relaxation around the SPP. This leads to the six-fold increase in the iron concentration observed in experiments. In the model, the loss of SPP/matrix coherency after irradiation causes the dislocations to drift away from the interface, and the iron concentration is homogenised accordingly. The hydrogen concentration was also predicted and found to be about 50% higher than in the bulk zirconium matrix at room temperature. The computational framework is built to be fast, making possible a statistical analysis on over five hundred simulations for improved reliability of the predictions.

A mechanistic understanding of hydrogen diffusion and hydride precipitation at the microscale underpins the prediction of delayed hydride cracking in zirconium alloy nuclear fuel cladding. We present a novel approach to modelling the microstructures created by hydride precipitation at loaded notches in polycrystalline Zr alloys. The model is multi-scale in that it includes the elastic dipole tensor of interstitial hydrogen in α-Zr, it treats the stress-driven diffusion of hydrogen at the meso-level (mm), it calculates the thermodynamically favourable spatial arrangement of microhydrides and their assembly into macrohydride colonies, in a textured polycrystalline sample, and it treats the full elastic field of the loaded notch and all the hydrides at a scale similar to the cladding thickness. A simplifying innovation is the representation of the elastic field of a microhydride by a dislocation dipole, where the Burgers vector is set to create the experimentally measured strain in the 〈11¯00〉 direction. The model provides a predictive framework for treating elastic anisotropy, a variety of potential nucleation sites, and different grain sizes. Simulated micrographs of hydride networks in polycrystalline samples with blunt and sharper loaded notches are compared with experimental micrographs obtained at the same scale. The simulations are extremely fast and calculations typically take around tens of seconds. This makes it possible to carry out detailed sensitivity studies with respect to several pertinent metallurgical variables, as well as conducting ensemble averaging of hydride microstructures.

This short book describes ten fundamental concepts – big ideas – of materials science. Some of them come from mainstream physics and chemistry, including thermodynamic stability and phase diagrams, symmetry, and quantum behaviour. Others are about restless atomic motion and thermal fluctuations, defects in crystalline materials as the agents of change in materials, nanoscience and nanotechnology, materials design and materials discovery, metamaterials, and biological matter as a material. A cornerstone of materials science is the idea that materials are complex systems that interact with their environments and display the emergence of new science from the collective behaviour of atoms and defects. Great attention is paid to the clarity of explanations using only high school algebra and quoting the occasional useful formula. Exceptionally, elementary calculus is used in the chapter on metamaterials. It is not a text-book, but it offers undergraduates and their teachers a unique overview and insight into materials science. It may also help graduates of other subjects to decide whether to study materials science at postgraduate level.

Abstract The elastostatic Green’s tensor function is the solution of a differential equation for the displacement field created by a unit point force in an infinite continuum. Its symmetry is derived using Maxwell’s reciprocity theorem. A general integral expression is derived for the Green’s function in anisotropic media. The Green’s function in isotropic elasticity is derived in closed form. The relation between the elastic Green’s function in a continuum and in a harmonic crystal lattice is shown. The application of the Green’s function to solving displacement fields of point defects exerting defect forces on neighbouring atoms leads to dipole, quadrupole, octupole, etc. tensors for point defects. Eshelby’s ellipsoidal inclusion problem is solved in isotropic elasticity. Using perturbation theory analytic expressions for the Green’s function in a weakly anisotropic cubic crystal are obtained in problem 3 of set 4. The derivation of the elastodynamic Green’s function in isotropic elasticity is outlined.

In a Volterra dislocation the relative displacement by the Burgers vector appears abruptly in the dislocation core so that the core has no width. This leads to divergent stresses and strains, which are unrealistic. Hybrid models correct this failure by considering a balance of forces that results in a finite core width, and finite stresses and strains throughout. Interatomic forces tend to constrict the core and elastic forces tend to widen it. The Frenkel-Kontorova model comprises two interacting linear chains of atoms as a representation of an edge dislocation, with linear springs between adjacent atoms of each chain. The Peierls-Nabarro model assumes the core is confined to two parallel atomic planes sandwiched between elastic continua. This model enables the stress to move the dislocation to be calculated, and it leads to the concept of dislocation kinks. These models highlight the role of atomic interactions in affecting ductility.

Understanding the plastic behaviour of thin zirconium hydrides is important for its implications on crack nucleation in the Zirconium alloy cladding used in fission reactors. Microvoids originate at fractured hydrides, and their coalescence may lead to the failure of the component. In this work, an innovative discrete dislocation framework is presented together with the preliminary results. The aim is to develop a model that is significantly faster than existing 3D formulations, to make it possible to run a statistical analysis on a simulated microstructure. This comes with limitations, which are discussed together with the planned developments. The model combines two planar and orthogonal simulations. In one only edge, in the other only screw dislocations are allowed, thereby describing all sides of a dislocation loop approximated as a rectangle. The two families of dislocations interact via their elastic stress, and this coupling proved to be important and significantly enhanced the dislocation density. The proposed model enables us to implement a 3D stress analysis of the hydrides. The simulations show that the most critical scenario is when neighbouring slip planes become populated with opposite-signed dislocations. This was observed both in the edge and in the screw case, and was reflected in the principal stress calculated by combining the two. It was also observed that the degree of permeability of the interface to dislocation crossing is inversely correlated to the stress inside the hydride and to the dislocation source activation.

David Pettifor was a theoretical physicist who changed the nature of materials science by raising the status of materials modelling to that of materials characterization and processing. He believed that the subject advanced through the development of simple models that withstood rigorous testing against experiments and the most accurate numerical computations. Having been a pioneer of total energy density functional theory calculations, he went on to derive analytic interatomic potentials for transition metals and nearly-free-electron metals and alloys from quantum mechanical principles. He is probably best known for the development of highly successful structure maps for binary and pseudo-binary alloys that were used by alloy developers in industry to create intermetallic alloys with improved properties. At Oxford he established the first materials modelling laboratory, bringing together physicists, chemists, materials scientists and engineers to model materials across length and time scales, which became a flagship laboratory for materials scientists world-wide.

A mathematical model for the embrittlement of a long elastic-plastic crack by a relatively small, misfitting inclusion is presented. The model makes direct contact with the Dugdale–Bilby–Cottrell–Swinden model as a limiting case. The particular case of an oxide inclusion with a triangular cross-section at the tip of an intergranular crack in the Ni-based superalloy RR1000 at \(650\,^{\circ }\hbox {C}\) is considered. The positive misfit of the intrusion provides an additional tensile load on the crack tip and on the plastic zone, raising the local stress intensity factor \(k_I\) and the crack tip opening displacement \({\varDelta } u\) above those when the inclusion is replaced by a dislocation-free zone of the same length. It is shown that for a given misfit strain and inclusion shape, the enhancement of \(k_I\) and \({\varDelta } u\) is controlled by a dimensionless parameter \(\omega = (\sigma /\sigma _1)\sqrt{c/(2l)}\) where \(\sigma \) is the applied stress, \(\sigma _1\) is the yield stress, c is the crack length and l is the length of the inclusion. The anti-shielding effect of the intrusion is significant only when \(\omega \lesssim 6\). As a result of the anti-shielding effect of the intrusion, the stress singularity at the crack tip always exceeds the compressive normal stress that exists within the thickest part of the intrusion when it is isolated. It is also shown that the gradient of the hydrostatic stress within the intrusion subjected to different applied stresses drives the oxygen diffusion and, hence, assists the oxidation at the grain boundary. The fracture toughness is considerably greater than that of a bulk sample of the oxide particle, which we attribute to the plastic zone.

This paper focuses on the study of the effect of the interfacial strength of grain boundaries and elliptical inclusions on crack path deflection. The method is developed to channel a crack into a toughening configuration (arrays of elliptical holes and inclusions are considered) in order to obtain the optimised microstructure required to enhance fracture toughness through different mechanisms. The proposed technique is shown to reproduce experimental crack propagation paths in various configurations and is capable of capturing the effect of that variation of the GB and the inclusion interfacial strength; it provides a powerful tool to understand the interplay between microstructural features and improve materials performance.

The Multipole Method (MPM) is used to simulate the many-body self-consistent problem of interacting elliptical micro-cracks and inclusions in single crystals. A criterion is employed to determine the crack propagation path based on the stress distribution; the evolution of individual micro-cracks and their interactions with existing cracks and inclusions is then predicted using what we coin the Discrete Crack Dynamics (DCD) method. DCD is fast (semi-analytical) and particularly suitable for the simulation of evolving low-speed crack networks in brittle or quasi-brittle materials. The method is validated against finite element analysis predictions and previously published experimental data.

Despite numerous theoretical models and simulation results, a clear physical picture of dislocations traveling at velocities comparable to the speed of sound in the medium remains elusive. Using two complementary atomistic methods to model uniformly moving screw dislocations, lattice dynamics and molecular dynamics, the existence of mechanical instabilities in the system is shown. These instabilities are found at material-dependent velocities far below the speed of sound. We show that these are the onset of an atomistic kinematic generation mechanism, which ultimately results in an avalanche of further dislocations. This homogeneous nucleation mechanism, observed but never fully explained before, is relevant in moderate and high strain rate phenomena including adiabatic shear banding, dynamic fracture, and shock loading. In principle, these mechanical instabilities do not prevent supersonic motion of dislocations.

The evolution of the defect microstructure in materials at high temperature is dominated by diffusion-mediated interactions between dislocations, cavities, and surfaces. This gives rise to complex nonlinear couplings between interstitial and vacancy-type dislocation loops, cavities, and the field of diffusing vacancies that adiabatically follows the evolution of microstructure. In our previous work, we developed a nonlocal model for the climb of curved dislocations and the morphological evolution of cavities during postirradiation annealing of structural components in nuclear reactors. We now expand the formalism to include the treatment of population of very small defects and dislocation loops that are below the experimental detection limit. These are taken into account through a mean field approach coupled with an explicit real-space treatment of larger-scale discrete defect clusters. We find that randomly distributed small defects screen diffusive interactions between larger discrete clusters, renormalizing the free diffusion Green's functions and transforming them into Yukawa-type propagators. The evolution of the coupled system is modelled self-consistently, showing how the defect microstructure evolves through a nonmonotonic variation of the distribution of sizes of dislocation loops and cavities, treated as discrete real-space objects.