Mark Kachanov | Tufts University (original) (raw)

Papers by Mark Kachanov

Encyclopedia of Continuum Mechanics, 2020

, Boston, USA) was a prominent scientist in the field of solid mechanics. His works address funda... more , Boston, USA) was a prominent scientist in the field of solid mechanics. His works address fundamental aspects of the field. He published several books distinguished by simplicity and clarity of presentation. One of them, "Fundamentals of the Theory of Plasticity," first published in Russian, has been translated into English (currently available as Dover publication), French, Chinese, and Japanese languages; it has been adopted as graduate-level textbook in many universities around the world.

Encyclopedia of Continuum Mechanics, 2020

Solid Mechanics and Its Applications, 2018

Formulation and solution of this problem by Eshelby [125-127] constitutes one of the major advanc... more Formulation and solution of this problem by Eshelby [125-127] constitutes one of the major advances in solid mechanics of the twentieth century. It has also led to revolutionary changes in mechanics of materials, by establishing a framework for quantitative modeling of phase transformations, effective properties of composites, stress concentrations at inhomogeneities, etc. It has been further advanced in a large number of works and constitutes the basic building block of micromechanics of materials. In accordance with needs of materials science and with the spirit of the book, the present chapter focuses on 3-D problems. Nevertheless, it also contains a section on 2-D problems, for the following two reasons. First, 2-D models may be relevant for certain applications such as fibers in composites or pores of pipe-like geometries occurring in geomaterials. Second, 2-D solutions allow analytic examination of various factors that cannot be done in 3-D (such as matrix anisotropy, or non-ellipsoidal inhomogeneity shapes). We emphasize, however, that such results provide only a qualitative guidance for the 3-D problems. We also discuss the inhomogeneity problem in the context of conductivity (electrical or thermal). The results are relevant for the effective conductivity of materials containing inhomogeneities, as well as full fields around the latter. These results can also be reformulated for other physical properties described by second-rank tensors such as the dielectric, transport, thermal expansion, and magnetic ones. 3.1 The First and the Second Eshelby Problems The name "Eshelby problem" actually covers two physically different problems (known as the first and the second Eshelby problems), given as follows:

WIT transactions on engineering sciences, 1998

Elastic interactions between holes of diverse eccentricities, as well as between holes and cracks... more Elastic interactions between holes of diverse eccentricities, as well as between holes and cracks, are analyzed. We examine various physical effects produced by mteracmons and the impact of hole eccentricities on these effects. Mixtures of holes of diverse eccentricities and sizes are of particular interest. Patterns of stress concentrations in such mixtures imply certain microfracturing patterns in materials with multiple defects. ^

WIT transactions on engineering sciences, 1998

We analyze some basic aspects of micromechanics of solids with multiple holes and microcracks. Tw... more We analyze some basic aspects of micromechanics of solids with multiple holes and microcracks. Two groups of properties are addressed: 1. Stress concentrations and microfracturing patterns in 2-D isotropic elastic solids with interacting elliptical holes. We examine various physical effects produced by interactions and the impact of hole eccentricities on these effects. Patterns of stress concentrations in such mixtures imply certain microfracturing patterns in materials with multiple defects. 2. Effective elastic properties of the isotropic and orthotropic solids with arbitrarily (randomly or non-randomly) oriented holes of diverse shapes. We describe the potential-based procedure to derive effective moduli of porous materials. The example of one family of parallel elliptical holes in the orthotropic matrix is considered in detail. 1 Interaction of holes in elastic solids To study interactions of elliptical holes (of diverse eccentricities) in a 2-D elastic solid, we employ the Neu...

WIT transactions on engineering sciences, 2000

Effective elastic moduli of a 2-D anisotropic solid with elliptical holes and cracks having an ar... more Effective elastic moduli of a 2-D anisotropic solid with elliptical holes and cracks having an arbitrary (non-random) orientational distribution are given in closed form. Proper tensorial parameters of defect density (dependent on ellipses' eccentricity and their orientations relative to the matrix anisotropy axes) are identified. This allows one to establish the overall elastic anisotropy. The results for mixtures of holes and cracks are presented.

CONTINUUM. MATHS. INFORMATICS. EDUCATION, 2020

Санкт-Петербургский государственный университет Аннотация. Собраны материалы по биографии Л.М. Ка... more Санкт-Петербургский государственный университет Аннотация. Собраны материалы по биографии Л.М. Качанова. Отмечается преданность учёного выбранного направления в науке, широта и глубина его исследований. Обсуждаются его методы обучения и руководства студентами и аспирантами. Ключевые слова: руководство кафедрой и научной работой студентов и аспирантов, преподавание основ механики деформируемого твёрдого тела, теории пластичности и ползучести, механики разрушения. Лазарь Маркович Качанов (1914-1993)-выдающийся советский механик, крупнейший специалист в области теории пластичности, ползучести, теории прочности и механики разрушения (рис.2).

International Journal of Engineering Science, 2019

We consider two elastic half-spaces contacting along multiple contact spots. In order to provide ... more We consider two elastic half-spaces contacting along multiple contact spots. In order to provide contact, these spots should be somewhat elevated; moreover, they may take the form of "columns" that connect the half-spaces. The effect of this factor on the overall compliances (normal and shear) is examined in 3-D setting, as function of the shape of the "columns". This effect is found to be strong, and it is quite sensitive to the mentioned shape. The extent of coupling between the shape of the "columns" and interactions between them, and of the elastic contrast between the "columns" and the main material are also examined.

International Journal of Engineering Science, 2018

We consider the effective elastic properties of cracked solids, and verify the hypothesis that th... more We consider the effective elastic properties of cracked solids, and verify the hypothesis that the effect of crack interactions on the overall anisotropy-its type and orientation-is negligible (even though the effect on the overall elastic constants may be strong), provided crack centers are located randomly. This hypothesis is confirmed by computational studies on large number of 2-D crack arrays of high crack density (up to 0.8) that are realizations of several orientation distributions. Therefore, the anisotropy can be accurately determined analytically in the non-interaction approximation (NIA). Since the effective elastic properties possess the orthotropic symmetry in the NIA (for any orientation distribution of cracks, including cases when, geometrically , the crack orientation pattern does not have this symmetry), the orthotropy of cracked solids is not affected by interactions.

International Journal of Solids and Structures, 2018

The compliance contribution of a penny-shaped crack having multiple contacts between crack faces ... more The compliance contribution of a penny-shaped crack having multiple contacts between crack faces is analyzed. In order to reduce the number of parameters involved, regular lattices of circular contacts are considered. It is found, in particular, that, at the same total area of contacts, a large number of small contacts produces stronger stiffening effect than a small number of large contacts; hence the total contact area is not the parameter controlling the crack compliance contribution. The effects of contacts on both the normal and the shear crack compliances are found to be very close. In the case of moderately nonregular contact lattices, the results were found to be only marginally different from the results for regular ones. Results are conveniently expressed in terms of "Holm's radius"-the radius of a single contact that would have produced the same effect. Being applied to multiple cracks with contacts, our results yield the "adjusted", for the presence of contacts, value of crack density.

Solid Mechanics and Its Applications, 2018

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

International Journal of Engineering Science, 2015

Explicit connection between conductance (electric or thermal) and incremental elastic stiffness o... more Explicit connection between conductance (electric or thermal) and incremental elastic stiffness of a rough contact of transversely-isotropic half-spaces is given. It applies to one or multiple contact areas of arbitrary geometries and extends the relation given earlier by Barber (2003) for the isotropic half-spaces. The dependence of the cross-property coefficient on the extent of the conductive anisotropy is relatively weak whereas the elastic anisotropy produces substantially stronger effect.

International Journal of Engineering Science, 2017

Philosophical Magazine, 2016

International Journal of Solids and Structures, 2002

Frictional sliding on a crack with non-uniform frictional characteristics is considered. The pres... more Frictional sliding on a crack with non-uniform frictional characteristics is considered. The present work continues the investigation of Gorbatikh et al. [Int. J. Solids Struct., in press] and focuses on the cyclic loading. The evolution of the sliding process in loading±reloading±unloading cycles is analyzed. We also extend the analysis to the important case when the frictional resistance changes in the process of sliding (such changes may model``degradation'' of the sliding surface during sliding, as well as other physical factors, not necessarily related to the sliding itself).

International Journal of Solids and Structures, 2001

JOURNAL OF VACUUM SCIENCE & TECHNOLOGY B, 2005

Physical Review B, 2007

A proper quantitative interpretation of scanning probe microscopy ͑SPM͒ experiments requires solu... more A proper quantitative interpretation of scanning probe microscopy ͑SPM͒ experiments requires solutions for both normal and tangential indentations of punches into a piezoelectric material. Such indentation solutions, their dependence on the indenter shape, and implications for SPM are considered here. More specifically, indentation of the spherical and conically sharp indenters into a piezoelectric half-space accompanied by frictional sliding is addressed. The tangential part of the problem, which involves friction, is solved to complement the solution of the normal indentation problem obtained earlier. Exact stiffness relations between vertical load, tangential displacement, and material properties are obtained. The piezoelectric coupling is found to have a relatively weak effect on lateral contact stiffness. In contrast, the contact area depends noticeably on the tangential effects. The full electroelastic fields are derived in elementary functions and their implications are discussed.

Physical Review B, 2004

To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolut... more To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolution limits, tip bias-and strain-induced phenomena and spectroscopy, analytical representations for tip-induced electroelastic fields inside the material are derived for the cases of weak and strong indentation. In the weak indentation case, electrostatic field distribution is calculated using an image charge model. In the strong indentation case, the solution of the coupled electroelastic problem for piezoelectric indentation is used to obtain the electric field and strain distribution in the ferroelectric material. This establishes a complete continuum mechanics description of the PFM contact mechanics and imaging mechanism. The electroelastic field distribution allows signal generation volume in PFM to be determined. These rigorous solutions are compared with the electrostatic point-charge and sphere-plane models, and the applicability limits for asymptotic point-charge and point-force models are established. The implications of these results for ferroelectric polarization switching processes are analyzed.

Encyclopedia of Continuum Mechanics, 2020

, Boston, USA) was a prominent scientist in the field of solid mechanics. His works address funda... more , Boston, USA) was a prominent scientist in the field of solid mechanics. His works address fundamental aspects of the field. He published several books distinguished by simplicity and clarity of presentation. One of them, "Fundamentals of the Theory of Plasticity," first published in Russian, has been translated into English (currently available as Dover publication), French, Chinese, and Japanese languages; it has been adopted as graduate-level textbook in many universities around the world.

Encyclopedia of Continuum Mechanics, 2020

Solid Mechanics and Its Applications, 2018

Formulation and solution of this problem by Eshelby [125-127] constitutes one of the major advanc... more Formulation and solution of this problem by Eshelby [125-127] constitutes one of the major advances in solid mechanics of the twentieth century. It has also led to revolutionary changes in mechanics of materials, by establishing a framework for quantitative modeling of phase transformations, effective properties of composites, stress concentrations at inhomogeneities, etc. It has been further advanced in a large number of works and constitutes the basic building block of micromechanics of materials. In accordance with needs of materials science and with the spirit of the book, the present chapter focuses on 3-D problems. Nevertheless, it also contains a section on 2-D problems, for the following two reasons. First, 2-D models may be relevant for certain applications such as fibers in composites or pores of pipe-like geometries occurring in geomaterials. Second, 2-D solutions allow analytic examination of various factors that cannot be done in 3-D (such as matrix anisotropy, or non-ellipsoidal inhomogeneity shapes). We emphasize, however, that such results provide only a qualitative guidance for the 3-D problems. We also discuss the inhomogeneity problem in the context of conductivity (electrical or thermal). The results are relevant for the effective conductivity of materials containing inhomogeneities, as well as full fields around the latter. These results can also be reformulated for other physical properties described by second-rank tensors such as the dielectric, transport, thermal expansion, and magnetic ones. 3.1 The First and the Second Eshelby Problems The name "Eshelby problem" actually covers two physically different problems (known as the first and the second Eshelby problems), given as follows:

WIT transactions on engineering sciences, 1998

Elastic interactions between holes of diverse eccentricities, as well as between holes and cracks... more Elastic interactions between holes of diverse eccentricities, as well as between holes and cracks, are analyzed. We examine various physical effects produced by mteracmons and the impact of hole eccentricities on these effects. Mixtures of holes of diverse eccentricities and sizes are of particular interest. Patterns of stress concentrations in such mixtures imply certain microfracturing patterns in materials with multiple defects. ^

WIT transactions on engineering sciences, 1998

We analyze some basic aspects of micromechanics of solids with multiple holes and microcracks. Tw... more We analyze some basic aspects of micromechanics of solids with multiple holes and microcracks. Two groups of properties are addressed: 1. Stress concentrations and microfracturing patterns in 2-D isotropic elastic solids with interacting elliptical holes. We examine various physical effects produced by interactions and the impact of hole eccentricities on these effects. Patterns of stress concentrations in such mixtures imply certain microfracturing patterns in materials with multiple defects. 2. Effective elastic properties of the isotropic and orthotropic solids with arbitrarily (randomly or non-randomly) oriented holes of diverse shapes. We describe the potential-based procedure to derive effective moduli of porous materials. The example of one family of parallel elliptical holes in the orthotropic matrix is considered in detail. 1 Interaction of holes in elastic solids To study interactions of elliptical holes (of diverse eccentricities) in a 2-D elastic solid, we employ the Neu...

WIT transactions on engineering sciences, 2000

Effective elastic moduli of a 2-D anisotropic solid with elliptical holes and cracks having an ar... more Effective elastic moduli of a 2-D anisotropic solid with elliptical holes and cracks having an arbitrary (non-random) orientational distribution are given in closed form. Proper tensorial parameters of defect density (dependent on ellipses' eccentricity and their orientations relative to the matrix anisotropy axes) are identified. This allows one to establish the overall elastic anisotropy. The results for mixtures of holes and cracks are presented.

CONTINUUM. MATHS. INFORMATICS. EDUCATION, 2020

Санкт-Петербургский государственный университет Аннотация. Собраны материалы по биографии Л.М. Ка... more Санкт-Петербургский государственный университет Аннотация. Собраны материалы по биографии Л.М. Качанова. Отмечается преданность учёного выбранного направления в науке, широта и глубина его исследований. Обсуждаются его методы обучения и руководства студентами и аспирантами. Ключевые слова: руководство кафедрой и научной работой студентов и аспирантов, преподавание основ механики деформируемого твёрдого тела, теории пластичности и ползучести, механики разрушения. Лазарь Маркович Качанов (1914-1993)-выдающийся советский механик, крупнейший специалист в области теории пластичности, ползучести, теории прочности и механики разрушения (рис.2).

International Journal of Engineering Science, 2019

We consider two elastic half-spaces contacting along multiple contact spots. In order to provide ... more We consider two elastic half-spaces contacting along multiple contact spots. In order to provide contact, these spots should be somewhat elevated; moreover, they may take the form of "columns" that connect the half-spaces. The effect of this factor on the overall compliances (normal and shear) is examined in 3-D setting, as function of the shape of the "columns". This effect is found to be strong, and it is quite sensitive to the mentioned shape. The extent of coupling between the shape of the "columns" and interactions between them, and of the elastic contrast between the "columns" and the main material are also examined.

International Journal of Engineering Science, 2018

We consider the effective elastic properties of cracked solids, and verify the hypothesis that th... more We consider the effective elastic properties of cracked solids, and verify the hypothesis that the effect of crack interactions on the overall anisotropy-its type and orientation-is negligible (even though the effect on the overall elastic constants may be strong), provided crack centers are located randomly. This hypothesis is confirmed by computational studies on large number of 2-D crack arrays of high crack density (up to 0.8) that are realizations of several orientation distributions. Therefore, the anisotropy can be accurately determined analytically in the non-interaction approximation (NIA). Since the effective elastic properties possess the orthotropic symmetry in the NIA (for any orientation distribution of cracks, including cases when, geometrically , the crack orientation pattern does not have this symmetry), the orthotropy of cracked solids is not affected by interactions.

International Journal of Solids and Structures, 2018

The compliance contribution of a penny-shaped crack having multiple contacts between crack faces ... more The compliance contribution of a penny-shaped crack having multiple contacts between crack faces is analyzed. In order to reduce the number of parameters involved, regular lattices of circular contacts are considered. It is found, in particular, that, at the same total area of contacts, a large number of small contacts produces stronger stiffening effect than a small number of large contacts; hence the total contact area is not the parameter controlling the crack compliance contribution. The effects of contacts on both the normal and the shear crack compliances are found to be very close. In the case of moderately nonregular contact lattices, the results were found to be only marginally different from the results for regular ones. Results are conveniently expressed in terms of "Holm's radius"-the radius of a single contact that would have produced the same effect. Being applied to multiple cracks with contacts, our results yield the "adjusted", for the presence of contacts, value of crack density.

Solid Mechanics and Its Applications, 2018

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

International Journal of Engineering Science, 2015

Explicit connection between conductance (electric or thermal) and incremental elastic stiffness o... more Explicit connection between conductance (electric or thermal) and incremental elastic stiffness of a rough contact of transversely-isotropic half-spaces is given. It applies to one or multiple contact areas of arbitrary geometries and extends the relation given earlier by Barber (2003) for the isotropic half-spaces. The dependence of the cross-property coefficient on the extent of the conductive anisotropy is relatively weak whereas the elastic anisotropy produces substantially stronger effect.

International Journal of Engineering Science, 2017

Philosophical Magazine, 2016

International Journal of Solids and Structures, 2002

Frictional sliding on a crack with non-uniform frictional characteristics is considered. The pres... more Frictional sliding on a crack with non-uniform frictional characteristics is considered. The present work continues the investigation of Gorbatikh et al. [Int. J. Solids Struct., in press] and focuses on the cyclic loading. The evolution of the sliding process in loading±reloading±unloading cycles is analyzed. We also extend the analysis to the important case when the frictional resistance changes in the process of sliding (such changes may model``degradation'' of the sliding surface during sliding, as well as other physical factors, not necessarily related to the sliding itself).

International Journal of Solids and Structures, 2001

JOURNAL OF VACUUM SCIENCE & TECHNOLOGY B, 2005

Physical Review B, 2007

A proper quantitative interpretation of scanning probe microscopy ͑SPM͒ experiments requires solu... more A proper quantitative interpretation of scanning probe microscopy ͑SPM͒ experiments requires solutions for both normal and tangential indentations of punches into a piezoelectric material. Such indentation solutions, their dependence on the indenter shape, and implications for SPM are considered here. More specifically, indentation of the spherical and conically sharp indenters into a piezoelectric half-space accompanied by frictional sliding is addressed. The tangential part of the problem, which involves friction, is solved to complement the solution of the normal indentation problem obtained earlier. Exact stiffness relations between vertical load, tangential displacement, and material properties are obtained. The piezoelectric coupling is found to have a relatively weak effect on lateral contact stiffness. In contrast, the contact area depends noticeably on the tangential effects. The full electroelastic fields are derived in elementary functions and their implications are discussed.

Physical Review B, 2004

To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolut... more To achieve quantitative interpretation of piezoresponse force microscopy (PFM), including resolution limits, tip bias-and strain-induced phenomena and spectroscopy, analytical representations for tip-induced electroelastic fields inside the material are derived for the cases of weak and strong indentation. In the weak indentation case, electrostatic field distribution is calculated using an image charge model. In the strong indentation case, the solution of the coupled electroelastic problem for piezoelectric indentation is used to obtain the electric field and strain distribution in the ferroelectric material. This establishes a complete continuum mechanics description of the PFM contact mechanics and imaging mechanism. The electroelastic field distribution allows signal generation volume in PFM to be determined. These rigorous solutions are compared with the electrostatic point-charge and sphere-plane models, and the applicability limits for asymptotic point-charge and point-force models are established. The implications of these results for ferroelectric polarization switching processes are analyzed.