The nonuniform motion of an edge dislocation in a singularly hyperbolic cubic (or hexagonal) crys... more The nonuniform motion of an edge dislocation in a singularly hyperbolic cubic (or hexagonal) crystal is analyzed, both for general nonuniform motion and for a motion starting from rest with constant velocity. In the latter case the solution is obtained in closed form from which the stress behaviour near the wavefront is also derived.
It is shown that a coupled mechanochemical system can be stable under no diffusion and unstable w... more It is shown that a coupled mechanochemical system can be stable under no diffusion and unstable with the addition of diffusion.
ABSTRACT A formulation is presented for the weight function of the wedge based on a finite variat... more ABSTRACT A formulation is presented for the weight function of the wedge based on a finite variation of the rate of change of virtual energy with an infinitesimal change with respect to a geometric variable, namely the angle of the wedge. ... 0 1. INTRODUCTION The weight ...
Abstract Sufficient conditions for instability of a nonlinear system exhibiting “negative creep” ... more Abstract Sufficient conditions for instability of a nonlinear system exhibiting “negative creep” (i.e. negative classical elastic moduli) are obtained on the basis of Hadamard analysis of the coupled system of partial differential equations. A coupled elastic system with diffusion only or energy and diffusion equations is analyzed and conditions on the moduli are obtained that render uniformly bounded initial data unstable.
When nuclei of strain approach the interface of two materials, the displacement fields may not be... more When nuclei of strain approach the interface of two materials, the displacement fields may not be unique and may depend on the direction from which the interface is approached. For example, the displacement fields of a center of dilatation at the interface of two materials are not unique and depend on the direction of approach to the interface. To avoid misunderstanding, it can be stressed that each of the two fields is continuous at the interface. In this paper, we show that there are 12 independent displacement functions of second-order singularities uniquely defined at an interface. The limits of all other nuclei of strain at the interface are linear combinations of these 12 independent displacement functions.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1998
By Xanthippi Markenscoff1 and Michael Paukshto2 1Department of Applied Mechanics, University of C... more By Xanthippi Markenscoff1 and Michael Paukshto2 1Department of Applied Mechanics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA 2Institute of Mathematics and Mechanics, Saint Petersburg State University, 198328, Box 125, St Petersburg, Russia
Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 2016
The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the ... more The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.
Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 2016
The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the ... more The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.
The M waves introduced by Burridge and Willis (1969) are emitted by the surface of a selfsimilarl... more The M waves introduced by Burridge and Willis (1969) are emitted by the surface of a selfsimilarly expanding elliptical crack, and they give Rayleigh waves at the corresponding crack speed. In the analysis for the self-similarly expanding spherical inclusion with phase change (dynamic Eshelby problem) the M waves are related to the waves obtained on the basis of the dynamic Green’s function containing the contribution from the latest wavelets emitted by the expanding boundary of phase discontinuity, and they satisfy the Hadamard jump conditions for compatibility and linear momentum across the moving phase boundary of discontinuity. In the interior of the expanding inclusion they create a “lacuna” with zero particle velocity by canceling the effect of the P and S. It is shown that the “lacuna” and Eshelby properties are also valid for a Newtonian fluid undergoing phase change in a self-similarly expanding ellipsoidal region of a fluid with different viscosity.
As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutio... more As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutions can be obtained both in statics and dynamics (for self-similarly expanding ones). Here is presented the detailed analysis of the static tension and shear cracks, as distributions of vertical centers of eigenstrains and centers of antisymmetric shear, respectively, inside the ellipse being flattened to a crack, so that the singular external field is obtained by the analysis, while the interior is zero. It is shown that a distribution of eigenstrains that produces a symmetric center of shear cannot produce a crack. A possible model for a Barenblatt type crack is proposed by the superposition of two elliptical inclusions by adjusting their small axis and strengths of eigenstrains so that the singularity cancels at the tip.
The coefficients of the 1 / ε 1/\varepsilon and ln ε \varepsilon singular terms in the field quan... more The coefficients of the 1 / ε 1/\varepsilon and ln ε \varepsilon singular terms in the field quantities near an arbitrarily moving dislocation loop are obtained by singular asymptotic expansion of integrals.
The nonuniform motion of an edge dislocation in a singularly hyperbolic cubic (or hexagonal) crys... more The nonuniform motion of an edge dislocation in a singularly hyperbolic cubic (or hexagonal) crystal is analyzed, both for general nonuniform motion and for a motion starting from rest with constant velocity. In the latter case the solution is obtained in closed form from which the stress behaviour near the wavefront is also derived.
It is shown that a coupled mechanochemical system can be stable under no diffusion and unstable w... more It is shown that a coupled mechanochemical system can be stable under no diffusion and unstable with the addition of diffusion.
ABSTRACT A formulation is presented for the weight function of the wedge based on a finite variat... more ABSTRACT A formulation is presented for the weight function of the wedge based on a finite variation of the rate of change of virtual energy with an infinitesimal change with respect to a geometric variable, namely the angle of the wedge. ... 0 1. INTRODUCTION The weight ...
Abstract Sufficient conditions for instability of a nonlinear system exhibiting “negative creep” ... more Abstract Sufficient conditions for instability of a nonlinear system exhibiting “negative creep” (i.e. negative classical elastic moduli) are obtained on the basis of Hadamard analysis of the coupled system of partial differential equations. A coupled elastic system with diffusion only or energy and diffusion equations is analyzed and conditions on the moduli are obtained that render uniformly bounded initial data unstable.
When nuclei of strain approach the interface of two materials, the displacement fields may not be... more When nuclei of strain approach the interface of two materials, the displacement fields may not be unique and may depend on the direction from which the interface is approached. For example, the displacement fields of a center of dilatation at the interface of two materials are not unique and depend on the direction of approach to the interface. To avoid misunderstanding, it can be stressed that each of the two fields is continuous at the interface. In this paper, we show that there are 12 independent displacement functions of second-order singularities uniquely defined at an interface. The limits of all other nuclei of strain at the interface are linear combinations of these 12 independent displacement functions.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1998
By Xanthippi Markenscoff1 and Michael Paukshto2 1Department of Applied Mechanics, University of C... more By Xanthippi Markenscoff1 and Michael Paukshto2 1Department of Applied Mechanics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA 2Institute of Mathematics and Mechanics, Saint Petersburg State University, 198328, Box 125, St Petersburg, Russia
Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 2016
The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the ... more The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.
Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 2016
The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the ... more The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.
The M waves introduced by Burridge and Willis (1969) are emitted by the surface of a selfsimilarl... more The M waves introduced by Burridge and Willis (1969) are emitted by the surface of a selfsimilarly expanding elliptical crack, and they give Rayleigh waves at the corresponding crack speed. In the analysis for the self-similarly expanding spherical inclusion with phase change (dynamic Eshelby problem) the M waves are related to the waves obtained on the basis of the dynamic Green’s function containing the contribution from the latest wavelets emitted by the expanding boundary of phase discontinuity, and they satisfy the Hadamard jump conditions for compatibility and linear momentum across the moving phase boundary of discontinuity. In the interior of the expanding inclusion they create a “lacuna” with zero particle velocity by canceling the effect of the P and S. It is shown that the “lacuna” and Eshelby properties are also valid for a Newtonian fluid undergoing phase change in a self-similarly expanding ellipsoidal region of a fluid with different viscosity.
As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutio... more As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutions can be obtained both in statics and dynamics (for self-similarly expanding ones). Here is presented the detailed analysis of the static tension and shear cracks, as distributions of vertical centers of eigenstrains and centers of antisymmetric shear, respectively, inside the ellipse being flattened to a crack, so that the singular external field is obtained by the analysis, while the interior is zero. It is shown that a distribution of eigenstrains that produces a symmetric center of shear cannot produce a crack. A possible model for a Barenblatt type crack is proposed by the superposition of two elliptical inclusions by adjusting their small axis and strengths of eigenstrains so that the singularity cancels at the tip.
The coefficients of the 1 / ε 1/\varepsilon and ln ε \varepsilon singular terms in the field quan... more The coefficients of the 1 / ε 1/\varepsilon and ln ε \varepsilon singular terms in the field quantities near an arbitrarily moving dislocation loop are obtained by singular asymptotic expansion of integrals.
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