Xanthippi Markenscoff

Abstract We say that bodies extended in space possess an ‘active’ microstructure when they have internal degrees of freedom with related peculiar interactions that cannot be completely described by the standard stress. These interactions may alter the material stability. Internal (observable) degrees of freedom are described by phase fields additional to the deformation. When they are linked to the macroscopic strain by an internal constraint, the emerging scheme reduces to higher-gradient extensions of standard continuum mechanics, which describe second-neighbour or higher-order interactions. These can strengthen or weaken the material stability. To explore these effects, comparing different but linked modeling approaches, we conduct a skeletal analysis referring to the Hadamard stability of an elastic one-dimensional body, described under varying assumptions about its internal structure. The analysis refers to the long wavelength with respect to the microstructural length. We show that in continua with active microstructure stability competes with multi-field destabilizing effects. Then, looking at a three-dimensional (3D) setting and general manifold-valued phase fields, we also extend to the multi-field setting a uniqueness theorem by Ericksen & Toupin on the displacement boundary value static problem of small deformations superposed on finite strains. While the mathematical treatment is intentionally rather elementary, it provides conceptual insights into the description of multi-phase materials with active microstructures.

A defect of phase change in density and change in moduli modeled as a self-similarly dynamically expanding Eshelby ellipsoidal inhomogeneous inclusion can nucleate and grow under high pressure at a critical loading. The self-similarly expanding ellipsoid possesses the “lacuna” property, the particle velocity vanishing in the interior domain, which allows the constant stress Eshelby property to be valid in the interior and the inclusion to grow as a whole in the presence of inertia. The energetics for nucleation and growth are derived from the energy-momentum tensor and first principles and generalize in the presence of inertia the “force on an interface” obtained in statics by Eshelby based on a thought experiment. The solution obtains the flow of energies across the moving phase boundary of an inhomogeneous inclusion, at the balancing of which, corresponding to the vanishing of the M integral, the interface presents no obstacle, and an arbitrarily small inclusion of phase change nucleates and grows at constant potential energy. The solution explains the generation of a shear seismic source radiation in deep-focus earthquakes and the generation of failure waves producing a zone with micro-fractures under compression in lima glass, and has wider applications to amorphization defects, defects in alloys, laser additive manufacturing, etc.

Abstract Deep‐focus earthquakes (DFEs) originating at the Mantle‐Transition‐Zone (MTZ) (400–700 km) have a Double Couple (DC) radiation pattern similar to crustal earthquakes; however, their mechanism is different and governed by high pressures (15–25 GPa) at nucleation depths. We present a model of nucleation and growth of regions of phase transformation, undergoing a sudden reduction in volume (5%–10%), “volume collapse.” Successive symmetry‐breaking instabilities minimize the energy spent to move the boundary of phase discontinuity and a collapsing volume expands as a flattened pancake‐like self‐similarly expanding Eshelby ellipsoidal inclusion. At the vanishing of the M integral, expressing the balance of flows of energy across the inclusion boundary, at a critical value of the pressure, an arbitrarily small inclusion nucleates and grows at constant potential energy driven by the pressure acting on the change in volume. The inclusion develops shear eigenstrains that decompose into two DC, placing one on the basal plane to radiate without energy losses. The symmetric volume collapse radiates out as an anti‐symmetric DC, and the radiated energy is obtained as the “excess energy,” of the ambient pressure acting on the “volume collapse,” reduced by the energy consumed for the growth of the pancake surfaces, with a “pressure drop” ( p 0 − p cr ) driving the expansion emitting a DC, even under full isotropy. The solution explains some features of the DFEs, (a) the DC radiation, (b) their large energies (the Mw 8.3 Okhotsk earthquake [2013]), (c) the absence of volumetric radiation, and (d) why they can originate in the MTZ, a long‐standing open problem.

In a seminal paper in the Philosophical Transactions of the Royal Society (A244, 87–112). Eshelby (Eshelby 1951 Phil. Trans. R. Soc. Lond. A 244, 87–112. ( doi:10.1098/rsta.1951.0016 )) introduced the concept of ‘the force on an elastic singularity’ and suggested that the extensions to dynamics include the application of the momentum flux. In this direction, it is shown that, in the non-uniformly motion of a dislocation there is a loss in quasi-momentum (or pseudo-momentum) across the scales ε to ε 2 , which induces an effective mass of the dislocation, and a loss in kinetic energy across the scales. It is shown through Noether's theorem that the rate of change of quasi-momentum in the volume enclosing the dislocation is equal to the flux of it through the surface minus a quasi-force, which is the dynamic J integral. The connection between the variation in the Hamiltonian and in the Lagrangian relates the quasi-force to the energy-release rate, yielding the same effective mass, while providing physical meaning both through momentum and energy. The effective mass of a dislocation is important in relating the energetics of defects from the micro to the macro scales, and the loss of quasi-momentum can have wider applications in continua. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Abstract The near-field logarithmic singularities in the field quantities associated with the acceleration of an arbitrarily moving edge dislocation are calculated based on a conservation law involving the dynamic energy-momentum tensor integrated over a domain enclosed by a multi-scale contour (an annulus of inner radius ϵ02 and outer radius ϵ0). The existence of the logarithmic singularities is obtained solely from the conservation law and the leading 1/r terms in the near fields of the stress and the velocity (which are those of the steady-state motion with velocity the instantaneous velocity in the accelerating motion). From the equations of motion and the symmetry in the second partial derivatives of the displacements for y≠0 we obtain that all six logarithmic terms of the near-field expansions are independent of the angle in the polar coordinates. All logarithmic terms in the near-field expansion of the strains and velocity in an arbitrarily moving edge dislocation (subsonically) are evaluated.

We present a methodology for the study of the fields (stress, strain, velocity, displacement) radiated from an arbitrarily accelerating dislocation (screw or edge) in an anisotropic solid. For general symmetry, the Laplace transforms in time and space lead to the solution of a sextic algebraic equation. The wave-front behavior is obtained by a geometric interpretation of the Cagniard-de Hoop contour (used for the inversion of the transforms).