# Micropolar Theory of Elasticity

Brulin 0., Hjalmars 8., Linear grade-consistent micropolar theory, Int. J. Eng. Sci., 1981, 19, 1731. Burchuladse T. V., Gegelya T.G., Development of the method of potentials in elasticity theory, Tbilisi, 1985, preprint, (in Russian). The monograph "Micropolar Theory of Elasticity" is devoted to the asymmetric theory of elasticity and thermoelasticity, aiming at researchers and postgraduate students in solid mechanics and applied mathematics, as well as mechanical engineers. It offers various new results including the basic field equations, general methods of integration of basic equations, formulations of problems, as well as solutions to particular problems. The presented general solutions cover those of Galerkin, Green-Lamé and Papkovitch-Neuber type, whereas the formulations include the displacement-rotation problems as well as pure stress problems of asymmetric elastodynamics. Solutions to stationary 3D and 2D problems for a half-space, and singular solutions to 3D and 2D asymmetric elastodynamics and the thermoelasto-dynamics problems for an infinite space are given.

# LINEAR THEORY OF MICROPOLAR ELASTICITY

The couple stress theory is shown to emanate as a spacial case of the present theory when the motion is constrained so that micro- and macro-rotations coincide. Several energy and uniqueness theorems are given. (Author). Equations of motion, constitutive equations and boundary conditions are presented for a special class of micro-elastic materials called Micropolar Solids. These solids respond to micro-rotational motions and spin inertia and can support couple stress and distributed body couples. The couple stress theory is shown to emanate as a spacial case of the present theory when the motion is constrained so that micro- and macro-rotations coincide. Several energy and uniqueness theorems are given. (Author).

# Foundations of Micropolar Mechanics

E. Reissner, A further note on the equations of finite-strain force and moment stress elasticity. ZAMP 38, 665–673 (1987) E.Kröner,(ed.) ... Theory of Micropolar Elasticity, CISM Courses and Lectures, vol. 25 (Springer, Wien, 1970) W. The book presents foundations of the micropolar continuum mechanics including a short but comprehensive introduction of stress and strain measures, derivation of motion equations and discussion of the difference between Cosserat and classical (Cauchy) continua, and the discussion of more specific problems related to the constitutive modeling, i.e. constitutive inequalities, symmetry groups, acceleration waves, etc.

# Stress Analysis by Boundary Element Methods

The linear micropolar theory of elasticity [1, 121] may be considered to be a special part of the general non-linear theory of micropolar continua [70, 72]. The linear theory of micropolar thermoelasticity was developed by Nowacki [117, ... The boundary element method is an extremely versatile and powerful tool of computational mechanics which has already become a popular alternative to the well established finite element method. This book presents a comprehensive and up-to-date treatise on the boundary element method (BEM) in its applications to various fields of continuum mechanics such as: elastostatics, elastodynamics, thermoelasticity, micropolar elasticity, elastoplasticity, viscoelasticity, theory of plates and stress analysis by hybrid methods. The fundamental solution of governing differential equations, integral representations of the displacement and temperature fields, regularized integral representations of the stress field and heat flux, boundary integral equations and boundary integro-differential equations are derived. Besides the mathematical foundations of the boundary integral method, the book deals with practical applications of this method. Most of the applications concentrate mainly on the computational problems of fracture mechanics. The method has been found to be very efficient in stress-intensity factor computations. Also included are developments made by the authors in the boundary integral formulation of thermoelasticity, micropolar elasticity, viscoelasticity, plate theory, hybrid method in elasticity and solution of crack problems. The solution of boundary-value problems of thermoelasticity and micropolar thermoelasticity is formulated for the first time as the solution of pure boundary problems. A new unified formulation of general crack problems is presented by integro-differential equations.

# Mechanics Of Micropolar Media

Eringen A.C., "Theory of Micropolar Fluids," J. Math. Mech. , 16 (1966) 1. --- Grioli G., “Elasticita as immetrica," Annal Matimatica Pura Applicata, Ser. 450, (1960) 389. Koster W.T., "Couple-stresses in the Theory of Elasticity," Proc ... This book is essentially made up of the lecture notes delivered by seven authors at the International Centre for Mechanical Sciences in Udine in June 1979. It attempts to provide an up-to-date and concise summary of the authors' understanding of micropolar materials. Both asymmetric elasticity and fluids are covered. The chapters range from the discussion of micropolar molecular models to the analysis of structure models, from linear to nonlinear theories and from electromagnetic, thermal, viscous effects to lattice defects. The subjects are treated from both theoretical and experimental points of view. Students with physics, mathematics and mechanical backgrounds as well as professionals will find this treatise useful for study and reference.

# Topics in Clifford Analysis

From practical point of view, the micropolar theory models not only displacements of a continuum, as in the classical theory of elasticity, but also its rotations. Therefore, the micropolar theory assures a more precise description of ... Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions. The unique starting point of Wolfgang Sprößig’s work was the application of quaternionic analysis to elliptic differential equations and boundary value problems. Over the years, Clifford analysis has become a broad-based theory with a variety of applications both inside and outside of mathematics, such as higher-dimensional function theory, algebraic structures, generalized polynomials, applications of elliptic boundary value problems, wavelets, image processing, numerical and discrete analysis. The aim of this volume is to provide an essential overview of modern topics in Clifford analysis, presented by specialists in the field, and to honor the valued contributions to Clifford analysis made by Wolfgang Sprößig throughout his career.

# Shell Structures Theory and Applications

Main problem of the general theory of micropolar elastic thin plates and shells is in approximate, but adequate reduction of three-dimensional boundary-value problem of the theory of micropolar elasticity to a twodimensional problem. Shells are basic structural elements of modern technology and everyday life. Examples are automobile bodies, water and oil tanks, pipelines, aircraft fuselages, nanotubes, graphene sheets or beer cans. Also nature is full of living shells such as leaves of trees, blooming flowers, seashells, cell membranes, the double helix of DNA or wings of insects. In the human body arteries, the shell of the eye, the diaphragm, the skin or the pericardium are all shells as well. Shell Structures: Theory and Applications, Volume 3 contains 137 contributions presented at the 10th Conference “Shell Structures: Theory and Applications” held October 16-18, 2013 in Gdansk, Poland. The papers cover a wide spectrum of scientific and engineering problems which are divided into seven broad groups: general lectures, theoretical modelling, stability, dynamics, bioshells, numerical analyses, and engineering design. The volume will be of interest to researchers and designers dealing with modelling and analyses of shell structures and thin-walled structural elements.

# THEORY OF MICROPOLAR ELASTICITY

The article presents a self-contained account of the recent theory of micropolar elasticity. The article presents a self-contained account of the recent theory of micropolar elasticity. Micropolar elastic materials possess extra independent degrees of freedom for the local rotations different from the rotations of the classical elasticity. These materials respond to spin inertia and body and surface couples, and as a consequence they exhibit certain new static and dynamic effects, e.g., new types of waves and couple stresses. Extensive discussions are presented on deformation, strain, microstrain, rotations, microrotations, kinematics, and balance laws. The thermodynamics of micropolar solids is formulated and the consequences of entropy and inequality are discussed. Field equations, boundary and initial conditions are obtained. The indeterminate couple stress theory is shown to result as a special case of the theory. Several static and dynamic problems are solved on the subjects of reflection of various types of micropolar waves, surface waves, stress concentration around a circular hole, and force and moment singularities in infinite solids. (Author).

# Local Gradient Theory for Dielectrics

1.3.2 Polar and Micropolar Electroelastic Continuum The classical theory of elasticity envisions a solid body as a continuum of material points, each with infinitesimal size and no inner structure. The motion of such points is described ... This book is devoted to the development of the local gradient theory of dielectrics. It presents a brief description of the known approaches to the construction of generalized (integral- and gradient-type) continuous theories of dielectrics. It describes a new continuum–thermodynamic approach to the construction of nonlinear high-order gradient theory of thermoelastic non-ferromagnetic polarized media. This approach is based on accounting for non-diffusive and non-convective mass fluxes associated with the changes in the material microstructure. Within the linear approximation, the theory has been applied to study transition modes of the formation of near-surface inhomogeneity of coupled fields in solids, disjoining pressure in thin films, etc. The theory describes a number of observable phenomena (including the surface, size, flexoelectric, pyroelectric, and thermopolarization effects in centrosymmetric crystals, the Meads anomaly, the high frequency dispersion of elastic waves, etc.) that cannot be explained within the framework of the classical theory of dielectrics.

# Micropolar Theory of Shells and Plates

1.1 Preliminary Remarks Here without going into details the necessary information for the future utilization of the theory of non-symmetric elasticity is presented. The chapter is written based mainly on well-known publications [1, 2]. For the first time, the Micropolar Theory of Elasticity is applied to solving a wide variety of problems connected to the specifics of nanomaterials. Namely, their unique physical-mechanical characteristics and behaviors under various stress-induced conditions. These theories have been constructed based on the equations of the classical theory of elasticity as well as other equations that have till now remained untouched in their application to molecular theories of solid deformable media. The book also introduces a new applied micropolar theory of thin shells which is based on Cosserat's pseudo-continuum. It explores the theory’s application to a category of nanomaterial shells and plates previously neglected from classical theories due to their unconventional size and structure. Theoretical results are accompanied by solutions of certain problems, essential for various applications. The book consists of six chapters. The first chapter is a review of the essential data on the non-symmetric theory of elasticity. The second and third chapters are devoted to various theories of plate bending and solutions to some basic problems. Chapter four refers to membrane or, so-called, momentary shell theory. Chapter five deals with the theory of very shallow shells. Finally, chapter six presents the geometry of the nonlinear theory of plates and the theory of very shallow shells. The book is intended for researchers, postgraduate students, and engineers, interested in the design of structures from nanomaterials and in the problems of mechanics of deformable bodies, theories of shells and plates, and their applications in micromechanics.

# Theory of Asymmetric Elasticity

NOWACKI , W. , On certain thermoelastic problems in micropolar elasticity , Bull . Acad . Polon . ... NOWACKI , W. , Distorsion problems in the micropolar theory of elasticity ( in Polish ) , Arch . Inż . Ląd . , 1972 , 18 , 3-4 , 375 . # Handbook of Research on Recent Developments in Electrical and Mechanical Engineering

Background Conventional linear elasticity theory successfully describe the man-made and naturally occurring materials ... For the in-depth knowledge of historical development of micropolar theory, authors would like to emphasize on some ... Technological advancements continue to enhance the field of engineering and have led to progress in branches that include electrical and mechanical engineering. These technologies have allowed for more sophisticated circuits and components while also advancing renewable energy initiatives. With increased growth in these fields, there is a need for a collection of research that details the variety of works being studied in our globalized world. The Handbook of Research on Recent Developments in Electrical and Mechanical Engineering is a pivotal reference source that discusses the latest advancements in these engineering fields. Featuring research on topics such as materials manufacturing, microwave photons, and wireless power transfer, this book is ideally designed for graduate students, researchers, engineers, manufacturing managers, and academicians seeking coverage on the works and experiences achieved in electrical and mechanical engineering.

# Recent Approaches in the Theory of Plates and Plate Like Structures

PergamonPress, Oxford Pal'mov V (1964) Fundamental equations of the theory of asymmetric elasticity. J Appl Math Mech 28(6):1341–1345 Pietraszkiewicz W, Eremeyev V (2009) On natural strain measures of the non-linear micropolar continuum ... This book presents the various approaches in establishment the basic equations of one- and two-dimensional structural elements. In addition, the boundaries of validity of the theories and the estimation of errors in approximate theories are given. Many contributions contain not only new theories, but also new applications, which makes the book interesting for researcher and graduate students.

# Foundations of Micropolar Thermoelasticity

The micropolar theory treated here relies heavily on the papers of Eringen and his coworkers. 4 Introduction linear Theory Résumé of Basic Equations of Linear Micropolar Thermoelasticity 93 Field Equations of the Linear Theory. 