Open questions

Four areas requiring further research are introduced and possible PhD projects are identified. They are (i) workhardening, (ii) electroplasticity, (iii) mobility of dislocations and (iv) hydrogen-assisted cracking. In each case the topic is introduced and key questions are identified. Self-organised criticality and slip bands are considered in the discussion of work hardening. The impact of drag forces is considered in the discussionof dislocation mobility. Possible mechanisms for hyfrogen-assisted cracking include hydrogen-enhanced decohesion (HEDE), adsorption-induced dislocation emission (AIDE) and hydrogen-enhanced localised plasticity (HELP).

Stress

The concept of stress is introduced in terms of interatomic forces acting through a plane, and in the Cauchy sense of a force per unit area on a plane in a continuum. Normal stresses and shear stresses are defined. Invariants of the stress tensor are derived and the von Mises shear stress is expressed in terms of them. The conditions for mechanical equilibrium in a continuum are derived, one of which leads to the stress tensor being symmetric. Stress is also shown to be the functional derivative of the elastic energy with respect to strain,which enables the stress tensor to be derived in models of interatomic forces. Adiabatic and isothermal stresses are distinguished thermodynamically and anharmonicity of atomic interactions is identified as the reason for their differences. Problems set 2 containsfour problems, one of which is based on Noll’s insightful analysis of stress and mechanical equilibrium.

The force on a defect

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.

Point defects

Examples of intrinsic and extrinsic point defects are discussed. Models of point defects in a continuum as misfitting spheres are solved for rigid and deformablemisfitting spheres. Free surfaces alter significantly the formation volume of a point defect even when the point defect is far from any free surface. Many point defects have non-sperical symmetry, and it is then better to consider defect forces exerted by the point defect on neighbouring atoms. Defect forces capture the symmetry of the point defect in its local environment. Interaction energies between point defects and between point defects and other sources of stress are expressed conveniently and with physical transparency in terms of dipole, quadrupole etc. tensors of point defects and derivatives of the Green’s function. The dipole tensor is experimentally measurable through the lambda-tensor, which measures the derivative of the macroscopic strain of a crystal with concentration of the point defect.

Hybrid models of dislocations

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.

Hooke’s law and elastic constants

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 three independent elastic constants in cubic crystalsare derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in non-cubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensorand compatibility stresses.

Strain

A discussion of the continuum approximation is followed by the definition of deformation as a transformation involving changes in separation between points within a continuum. This leads to the mathematical definition of the deformation tensor. The introduction of the displacement vector and its gradient leads to the definition of the strain tensor. The linear elastic strain tensor involves an approximation in which gradients of the displacement vector are assumed to be small. The deformation tensor can be written as the sum of syymetric and antisymmetric parts, the former being the strain tensor. Normal and shear strains are distinguished. Problems set 1 introduces the strain ellipsoid, the invariance of the trace of the strain tensor, proof that the strain tensor satisfies the transformation law of second rank tensors and a general expression for the change in separation of points within a continuum subjected to a homogeneous strain.

Cracks

Loaded slit cracks are modelled as continuous distributions of dislocations with infinitesimal Burgers vectors. Cauchy-type singular integral equations for the density of Burgers vector in these distributions are solved using the theory of Chebyshev polynomials. The elastic fields of mode I elastic slit cracks are derived and the stress intensity factor is defined. Other defects may interact with cracks such as dislocations. This leads to the concepts of shielding and anti-shielding of cracks by dislocations. The Dugdale–Bilby–Cottrell–Swinden model of a mode I crack completely shielded by a plastic zone is derived. By introducing a dislocation free zone between the plastic zone and the crack tip the crack tip is only partially shielded, enabling more brittle tendencies to be described. Griffith’s energy criterion for the growth of an existing crack is seen as necessary but not sufficient. The Barenblatt crack introduces the influence of interatomic forces at the crack tip.

Dislocations

Plastic deformation involves planes of atoms sliding over each other. The sliding happens through the movement of linear defects called dislocations. The phenomenology of dislocations and their characterisation by the Burgers circuit and line direction are described. The Green’s function plays a central role in Volterra’s formula for the displacement field of a dislocation and Mura’s formula for the strain and stress fields. The isotropic elastic fields of edge and screw dislocations are derived. The field of an infinitesimal dislocation loop and its dipole tensor are also derived. The elastic energy of interaction between a dislocation and another source of stress is derived, and leads to force on a dislocation. The elastic energy of a dislocation and the Frank-Read source of dislocations are also discussed. Problem set 6 extends the content of the chapter in several directions including grain boundaries and faults.

The Green’s function in linear elasticity

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.