Theory of elasticity

This paper is concerned with the existence, uniqueness and/or multiplicity, and stability of positive solutions of an indefinite weight elliptic problem with concave or convex nonlinearity. We use mainly bifurcation methods to obtain our results.

In this study, exact trigonometric displacement function was used to solve the buckling problem of a three-dimensional (3-D) rectangular plate that is clamped at the first-three edges and the other remaining edge simply supported (CCCS) under uniaxial compressive load. Employing 3-D constitutive relations which consist of entire components, the functional for total potential energy was obtained. After that, the rotation and deflection at x-axis and y-axis were formulated from the established compatibility equations to get an exact trigonometric deflection function. The characteristics equation was obtained by differentiating energy equation with respect to deflect to obtain the relations between deflection and rotation. The equation of the total potential energy is minimized with respect to the deflection coefficient after incorporating the deflection and rotation function, the critical buckling load formula was established. The solution for the buckling problem gotten shown that the structure of the plate is safe when the plate thickness is increased as the outcome of the study showed that the critical buckling load increased as the span-thickness ratio increased. The overall difference in form of percent between the present work and previous studies recorded is 5.4%. This shows that at about 95% certainty, the present work is perfect. The comparison of this study with the results of previous similar studies revealed the uniformity 3-D plate theory and the variations of CPT and RPT theories in the exact buckling analysis of a rectangular plate. However, this approach which includes all the six stress elements of the plate material in the analysis produced an exact deflection function unlike the previous studies which used assumed functions. Furthermore, the theoretical analysis of this study demonstrates a novel approach to solve the buckling problem rectangular plate which is capable of analyzing rectangular plates of any thickness configuration.

Solution of a problem of elasticity requires the solution of a system of 15 equations with 15 unknowns. A symmetric elimination of some the unknowns has allowed us to reduce the system of equations, either in terms of displacement or in terms of stresses. This suggests various methods that can be used in solving elasticity problems. Some of these methods depend primarily on intuition, while other are based on a systematic application of techniques of applied mathematics. This paper focuses on the numerical method for solution of two-dimensional elasticity problem.

Silicon rubbers are widely used in a variety of products ranging from cooking utensils and electronics to medical devices and implants. Recently, they have sparked an interest among soft robotics researchers as they can be easily formed into various shapes and actuated in a relatively fast and easy way. In this article, we examine the nonlinear elastic response of a silicon rubber, Ecoflex, under both compressible and incompressible constraints. An experimental test on a uniaxial tension indicates a slight compressibility, and the compressibility increases with stretching. Five different constitutive material models are considered to describe the nonlinear elastic responses of Ecoflex under both compressible and incompressible conditions. In addition, finite element (FE) analysis is presented to analyze multiaxial response of structures or devices made of Ecoflex under complex boundary conditions. This study highlights the variations in the multiaxial response of structures at large deformations from different constitutive models under different compressible and incompressible constraints. For a high precision control in soft robotics applications, there is a need to understand the multiaxial response of silicon rubbers, especially under large deformations.

W szczegółowy sposób omówione zostały założenia ogólnej nieliniowej teorii sprężystości i wyprowadzenia podstawowych elementów mechaniki ośrodka ciągłego, a także przytoczone zostały dowody fundamentalnych twierdzeń rozważanej teorii w ramach kinematyki, statyki i dynamiki ciał odkształcalnych. Omówiono kwestie formułowania związków konstytutywnych dla ciał izotropowych i anizotropowych zarówno o liniowej, jak i nieliniowej charakterystyce, ze szczególnym uwzględnieniem zagadnienia energetycznego sprzężenia miar naprężenia i odkształcenia oraz zastosowania rozkładu widmowego tensorów sprężystości. Osobno omówiono liniowy wariant teorii z różnymi jego sformułowaniami oraz metodami analitycznymi, m.in. zastosowaniem funkcji naprężeń, potencjałów, metod półodwrotnych oraz analizy zespolonej. Rozważania te poszerzono o opis teorii ogólniejszych – mechanikę ośrodka mikropolarnego Cosseratów oraz mechanikę ośrodka mikromorficznego Mindlina. W ramach teorii liniowej przedstawiono matematyczny opis zjawisk elastooptyki, propagacji płaskich i sferycznych fal sprężystych, a także propagacji sprężystych fal powierzchniowych. Przytoczono i udowodniono twierdzenia wariacyjne teorii liniowej oraz omówiono ich zastosowanie w ramach numerycznych metod rozwiązywania zagadnień tej teorii. Przeanalizowano rozwiązania trzynastu klasycznych zagadnień liniowej teorii sprężystości oraz zagadnień pokrewnych, ze szczególnym uwzględnieniem zagadnień dynamicznych. Całość uzupełniona została rozbudowanym dodatkiem poświęconym rachunkowi tensorowemu i absolutnemu rachunkowi różniczkowemu.

Tobacco industry has been transformed from being the industry with the highest profit margins to a potentially dead industry (Yuk & Wells, 2018). In 2018 share prices of all tobacco companies were shocked as investors lost faith in the profitability of future gains in tobacco industry. This has led some economists to state that tobacco companies should destroy themselves in order to survive (Badkar & Nicolaou, 2018). Worldwide market leader Philip Morris International has launched the new mission of “Designing a smoke – free future” declaring that it aims to “stop selling cigarettes “as soon as possible” (PMI, 2018).
The paper explores tobacco industry and PMI dynamics using microeconomics most relevant concepts.

A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals.

This paper deals with a finite-difference solution of the torsion problem of nonhomogeneous and compound prismatic bars. General, governing equations for both problems are developed and the boundary conditions for an interface between parts composed of homogeneous but different materials are stated. The case of multiply connected regions is discussed and integral conditions, analogous to the conditions in multiply connected homogeneous bars, are developed.

We investigated a high-purity cold-rolled martensitic Fe-9wt%Mn alloy. Tensile tests performed at room temperature after tempering for 6 h at 450 C showed discontinuous yielding. Such static strain ageing phenomena in Fe are usually associated with the segregation of interstitial elements such as C or N to dislocations. Here we show by correlative transmission electron microscopy (TEM)/atom probe tomog-raphy (APT) experiments that in this case Mn segregation to edge dislocations associated with the formation of confined austenitic states causes similar effects. The local chemical composition at the dislocation cores was investigated for different tempering temperatures by APT relative to the adjacent bcc matrix. In all cases the Mn partitioning to the dislocation core regions matches to the one between ferrite and austenite in thermodynamic equilibrium at the corresponding tempering temperature. Although a stable structural and chemical confined austenitic state has formed at the dislocation cores these regions do not grow further even upon prolonged tempering. Simulation reveals that the high Mn enrichment along the edge dislocation lines (25 at.%Mn at 450 C) cannot be described merely as a Cottrell atmosphere formed by segregation driven by size interaction. Thermodynamic calculations based on a multiscale model indicate that these austenite states at the dislocation cores are subcritical and defect-stabilized by the compression stress field of the edge dislocations. Phenomenologically, these states are the 1D equivalent to the so-called complexions which have been extensively reported to be present at 2D defects, hence have been named linear complexions.

"This note utilizes the analog models of forced vibration to express the mass of foundation block in terms of damping ratio and thereafter obtains the expression for the minimum foundation mass required to limit vertical machine vibration amplitude to a prescribed limit. The resulting formula, which accounts for internal damping, is constrained by the limitations of the original analog solutions.  Moreover, formulae are derived for determining the damping ratio for conditions different from optimum that enable the evaluation of the non-optimum foundation mass required to limit vibration. Since the ideal optimum solution may not be practical, expression has been derived for the best non-optimum
foundation mass. The method presented in this paper is believed to be easier to implement and to yield smaller foundation blocks than the existing traditional method, and thus will enable cheaper machine foundations to be constructed. It is believed that the proposed optimal solution may lead to the near elimination of machine vibration nuisance produced by high-frequency machines during operation.
"

Praca doktorska poświęcona opracowaniu kryterium stanu granicznego dla anizotropowych materiałów liniowo-sprężystych wykazujących różnicę wielkości naprężenia granicznego w zależności od jego znaku. Zaproponowane kryterium bazuje na koncepcji Rychlewskiego rozkładu głównego energii sprężystej oraz koncepcji Burzyńskiego wykorzystania zależnych od aktualnego stanu naprężenia funkcji skalujących kombinację elementów tego rozkładu. Zebrano i omówiono klasyczne kryteria dla materiałów anizotropowych oraz przeanalizowano i porównano tzw. "kryteria energetyczne". Przeprowadzono dowód twierdzenia Rychlewskiego dla wielowymiarowej przestrzeni stanów bezpiecznych. Szczegółowo scharakteryzowano poszczególne przypadki symetrii sprężystych oraz odpowiadające im postacie kryterium stanu granicznego.

The paper shows how a generalization of the elasticity theory to four dimensions and to space-time allows for a consistent description of the homogeneous and isotropic universe, including the accelerated expansion. The analogy is manifested by the inclusion in the traditional Lagrangian of general relativity of an additional term accounting for the strain induced in the manifold (i.e. in space-time) by the curvature, be it induced by the presence of a texture defect or by a matter/energy distribution. The additional term is sufficient to account for various observed features of the universe and to give a simple interpretation for the so called dark energy. Then, we show how the same approach can be adopted back in three dimensions to obtain the equilibrium configuration of a given solid subject to strain induced by defects or applied forces. Finally, it is shown how concepts coming from the familiar elasticity theory can inspire new approaches to cosmology and in return how methods appropriated to General Relativity can be applied back to classical problems of elastic deformations in three dimensions.

The classical theory suggests that phantom polymer networks have a treelike structure. In this study, we investigate the actual polymer networks, in which the polymer loops have a finite size and strongly overlap with each other. We show that the elastic modulus of such networks has two contributions: of elastically effective strands and of finite size loops shunting the network strands. The latter contribution substantially depends on the interaction of the network strand monomers at network preparation conditions. The result of calculations performed in the framework of the replica theory of polymer networks is interpreted using a generalized combined chain model. This model allows us to describe quantitatively the elasticity of the polymer network with finite size loops and the deformation of its individual strands. We also calculate the impact of primary loops and cyclic defects of arbitrary concentration on the elasticity of such a network.

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In this paper is presented the solution for an elastic problem for plane stress. The problem refers to a composite (built-up) plate from rectangular and centric circular ring, subjected to uniform external and internal pressure. The used method is the complex variable method and conformal mapping. The complex stress function can be found in the Laurenrt series and can be determined from the boundary conditions by solution a set of linear algebraic equations. The number of equations needed for sufficient accuracy depends upon the boundary shapes and loading conditions of the problem.

"Se presenta el análisis no lineal dinámico en tres direcciones de 6 edificiosde concreto reforzado de 5, 12 y 20 pisos, diseñados con la NSR-10 yla zonificación de respuesta sísmica de Bogotá del año 2010. Estos 6 edificios,con planta típica idéntica, fueron sometidos a 78 señales sísmicas del sismo deQuetame (año 2008) registradas por 26 estaciones de la red de acelerógrafosde Bogotá (RAB). Para el diseño de los edificios se utilizaron los espectros derespuesta de aceleración absoluta para las zonas: «Piedemonte-B» y «Lacustre-500» de la microzonificación sísmica de Bogotá. Posteriormente, se calcularonlas rótulas plásticas para cada elemento estructural de los 6 edificios, y en elcaso de las columnas, también sus diagramas de interacción. Luego, las edificacion estridimensionales fueron sometidas simultáneamente a los registros contra el tiempo norte-sur, este-oeste y vertical. Para cada edificio sometidoa las señales de cada estación, se obtuvieron las demandas de deriva, los desplazamientosde la cubierta, el cortante en la base y los niveles de daño segúnestimadores internacionalmente aceptados. Finalmente, se generaron mapasde demandas de deriva y aceleración por edificio en la ciudad de Bogotá D.C.Los resultados sugieren que los edificios diseñados con la amenaza sísmica dela nueva microzonificación de Bogotá y con las estipulaciones de la NSR-10,presentarán daños excesivos asociados a estados límites como son los de «seguridadde vidas» para un sismo de baja magnitud como fue el de Quetame,que equivalió a solo una fracción de aquel que corresponde al comportamientoreal bajo el escenario de diseño"

En este artículo se presenta un estado del arte donde aparecen los avances teóricos y empíricos que ha tenido el Sistema Casi Ideal de Demanda y, adicionalmente, las principales revistas académicas donde se ha publicado sobre el tema y los sectores donde se ha aplicado. Este trabajo está dividido en tres secciones: un panorama general, algunos avances teóricos y resultados teóricos sobre algunas estimaciones. Finalmente, se presentan unas conclusiones donde se expresa la importancia que tiene el modelo gracias a su versatilidad y facilidad para incluir variables no económicas en la explicación del comportamiento de la demanda.

The theory of elasticity predicts a variety of phenomena associated with solids that possess a negative Poisson&amp;#x27;s ratio. The fabrication of metamaterials with a ‘designed’ microstructure that exhibit a Poisson&amp;#x27;s ratio approaching the thermodynamic limits of 1/2 and −1 increases the likelihood of realising these phenomena for applications. In this work, we investigate the properties of a layered composite, with alternating layers of materials with negative and positive Poisson&amp;#x27;s ratio approaching the thermodynamic limits. Using the finite element method to simulate uniaxial loading and indentation of a free standing composite, we observed an increase in the resistance to mechanical deformation above the average value of the two materials. Even though the greatest increase in stiffness is gained as the thermodynamic limits are approached, a significant amount of added stiffness can be attained, provided that the Young&amp;#x27;s modulus of the negative Poisson&amp;#x27;s ratio material is not less than that of the positive Poisson&amp;#x27;s ratio material.

An elastic continuum approach for modeling the nonlinear vibration of a double-walled carbon nanotubes under harmonic excitation is presented. The carbon nanotube is modeled as two doubly clamped beams coupled through nonlinear continuous springs representing Van der Waal bonds. Geometric nonlinearity is included due to mid-plane stretching. A Galerkin approach is used to discretize the integropartial differential equation leading to two nonlinear coupled second-order ordinary differential governing equations. The numerically obtained dynamic response to a primary-resonance exciting the co-axially first and second vibration modes is investigated through frequency response, periodic and chaotic motions characteristics.

The wide application of switched reluctance motors (SRM) is limited by their higher vibration and acoustic noise. An accurate determination of resonant frequencies and mode shapes of the SRM stator is therefore essential for low acoustic noise design. Based on the structural finite element method and elasticity theory, 3D vibration modes and resonant frequencies of a 5 hp SRM with 8/6 poles are demonstrated and calculated in this paper. The stator stack with both a smooth frame and a ribbed frame are examined. The effects of an encased frame and a keyed frame on the free vibration modes and resonant frequencies of SRM stator are compared. Accelerometer test results on a SRM verify some of the numerical computations.

The problem of sound transmission through layered panel structures is studied with the exact theory of three-dimensional (3-D) elasticity. The exact solution to the 3-D elasticity equations is obtained by the use of the Fourier spectral method. Based on this analytical solution, a transfer matrix is derived that relates the spectral displacements and stresses on the one surface of the panel to those on the opposite panel surface. The transfer matrix is then used to develop the analytical solutions for sound reflection and transmission coefficients. Explicit, concise expressions are obtained for the analytical solutions of the acoustic transmission and reflection coefficients under the general conditions of layered anisotropic panels. Examples are given for both single-layer and sandwich panels. Predictions on sound transmission from the 3-D elasticity theory are compared with available data from other methods, and the results are discussed.

We propose a cloak for coupled shear and pressure waves in solids. Its elastic properties are deduced from a geometric transform that retains the form of Navier equations. The spherical shell is made of an anisotropic and heterogeneous medium described by an elasticity tensor C (without the minor symmetries) which has 21 non-zero spatially varying coefficients in spherical coordinates. Although some entries of C , e.g. some with a radial subscript, and the density (a scalar radial function) vanish on the inner boundary of the cloak, this metamaterial exhibits less singularities than its cylindrical counterpart studied in [M. Brun, S. Guenneau, A.B. Movchan, Appl. Phys. Lett. 94, 061903 (2009).] In the latter work, C suffered some infinite entries, unlike in our case. Finite element computations confirm that elastic waves are smoothly detoured around a spherical void without reflection.

The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the halfspace. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic halfspace deduced from Michell's expressions. A surprising conclusion is arrived at, namely: there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic halfspace. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of theory of elasticity.

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.

A third-order Hermitian zig-zag plate theory is presented as development of the classical cubic zig-zag displacement field. In addition to the capabilities of the previous model ((i) transverse shear flexibility, (ii) through-the-thickness continuity of the transverse shear stresses, (iii) traction-free condition on the two external surfaces of the laminate and (iv) possibility to study damaged interfaces), the Hermitian model offers some interesting improvements ((i) through-the-thickness linear varying transverse displacement, (ii) evaluation of the normal transverse deformability in general and of the corresponding normal stress in particular, (iii) traction equilibrium condition on the external surfaces and (iv) use of the displacements and transverse shear stresses of the external surfaces as degrees of freedom of the plate model). By means of the virtual work principle of the three-dimensional linear elasticity theory, the two-dimensional equations of motion and boundary conditions are obtained. Some numerical results are finally presented to show the particular nature of the through-the-thickness Hermitian shape functions and to test the model performances in evaluating the transverse normal stress.

A novel three-parameter, elastic foundation model is proposed in this study to analyze interface stresses of adhesively bonded joints. The classical two-parameter, elastic foundation model of adhesive joints models the adhesive layer as a layer of normal and a layer of shear springs. This model does not satisfy the zero-shear-stress boundary conditions at the free edges of the adhesive layer due to the inherent flaw of the two-parameter, elastic foundation model, which violates the equilibrium condition of the adhesive layer. To eliminate this flaw, this study models the adhesive layer as two normal spring layers interconnected by a shear layer. This new three-parameter, elastic foundation model allows the peel stresses along the two adherend/adhesive interfaces of the joint to be different, and therefore, satisfies the equilibrium condition of the adhesive layer. This model regains the missing “degree of freedom” in the two-parameter, elastic foundation model of the adhesive layer by introducing the transverse displacement of the adhesive layer as a new independent parameter. Explicit closed-form expressions of interface stresses and beam forces are obtained. The new model not only satisfies all boundary conditions, but also predicts correctly which interface has the strongest stress concentration. The new model is verified by continuum models existing in the literature and finite element analysis. The new three-parameter, elastic foundation model provides an effective and efficient tool for analysis and design of general adhesive joints.

Mathematical problems of the theory of steady state oscillations of hemitropic (chiral) elastic materials are considered. In the case of unbounded domains, the generalized Sommerfeld-Kupradze type radiation conditions are introduced and in the space of radiating solutions the uniqueness results are established. Applying the potential method and the theory of pseudodifferential equations, the unique solvability in various function spaces of the Dirichlet, Neumann and mixed boundary value problems for the steady state oscillation equations are proved. Regularity properties and representability of solutions by layer potentials are analyzed in the cases of smooth and Lipschitz domains. 2000 AMS Mathematics Subject Classification. Primary 35J55, 74H20, 74G25, 74G30.

Elastic capsules, prepared from droplets or bubbles attached to a capillary (as in a pendant drop tensiometer), can be deflated by suction through the capillary. We study this deflation and show that a combined analysis of the shape and wrinkling characteristics enables us to determine the elastic properties in situ. Shape contours are analyzed and fitted using shape equations derived from nonlinear membrane-shell theory to give the elastic modulus, Poisson ratio and stress distribution of the membrane. We include wrinkles, which generically form upon deflation, within the shape analysis. Measuring the wavelength of wrinkles and using the calculated stress distribution gives the bending stiffness of the membrane. We compare this method with previous approaches using the Laplace-Young equation and illustrate the method on two very different capsule materials: polymerized octadecyltrichlorosilane (OTS) capsules and hydrophobin (HFBII) coated bubbles. Our results are in agreement with the available rheological data. For hydrophobin coated bubbles the method reveals an interesting nonlinear behavior consistent with the hydrophobin molecules having a rigid core surrounded by a softer shell.

In this second paper of a series on the effective elastic properties of alumina-zirconia composite ceramics, principles of micromechnical modeling are reviewed and the most important relations are recalled. Rigorous bounds (Voigt-Reuss bounds) are given for the (scalar) effective elastic moduli (tensile modulus E, shear modulus G and bulk modulus K) of polycrystalline ceramics as calculated from monocrystal data (i.e. components of the elasticity tensor). Voigt-Reuss bounds and Hashin-Shtrikman bounds of the elastic moduli are given for two-phase composites. For porous materials, which can be considered as a degenerate case of two-phase composites where one phase is the void phase (with zero elastic moduli), micromechanical approximations (so-called dilute approximations, Dewey-Mackenzie formulae) are given. Apart from a heuristic extension of the dilute approximations in the form of so-called Coble-Kingery relations, semi-empirical extensions of the micromechanical approximations are given for the tensile modulus (Spriggs relation, modified exponential and Mooney-type relations, generalized / Archie-type power law relation, Phani-Niyogi / Krieger-type power law relation, Hasselman relation), including the new relation E/E 0 = (1φ) • (1φ/φ C), recently proposed by the authors, where E is the effective tensile modulus, φ the porosity, E 0 the tensile modulus of the dense (i.e. pore-free) ceramic material and φ C the critical porosity. Effective elastic properties of alumina-zirconia composite ceramics-Part 2. Micromechanical modeling Ceramics − Silikáty 48 (1) 14-23 (2004) Effective elastic properties of alumina-zirconia composite ceramics-Part 2. Micromechanical modeling Ceramics − Silikáty 48 (1) 14-23 (2004) Effective elastic properties of alumina-zirconia composite ceramics-Part 2. Micromechanical modeling

In this paper, we have given numerical solution of the elasticity problem of settled on the wronkler ground with variable coefficient. First we calculate the basic successive substitute approximation of the given boundary value problem then transform it into Pad e series form, which give an arbitrary order for solving differential equation numerically. We applied these methods to an example which is the elasticity problem of unit length homogeny beam, which is a special form of boundary value problem.

A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals.

The scope of the present article is the study of the fracture initiation from initial spherical flaws in incompressible propellant materials. Using a constitutive law, derived from the neo-Hookean strain energy function, one may deduce the relation between the radial pressure applied at infinity and the corresponding extension ratio, at the surface of the cavity. An equation which defines the critical extension ratio for a given value of a parameter, κ, was derived and the corresponding critical load, pc, is related to the critical cavity extension ratio. Plots for the extension ratio and normalized pressure were provided solving the appropriate equation derived from the analysis. The failure of the cavity is associated with the incremental energy transfer at a constant critical stress between the applied work, internal strain energy and surface energy. The influence of the surface energy on the radial deformation of an initially intact sphere is presented and it turns out that a characteristic cavitation phenomenon occurs instead of the bifurcation. A mathematical formula is derived for the pressure, per unit current area, which is valid for a general form of incompressible elastic strain energy function. Using the derived formula one may deduce the relation between the radial pressure and the corresponding extension ratio at the surface of the cavity. In an incompressible material of given modulus and surface energy it is proved that a critical field stress exists at which a small internal cavity may grow in size where the significant quantities are the radial and tangential strains.

The fundamental equations of the isotropic and anisotropic plane theory of elasticity are reworked in the frame of the generalized analytic functions theory. The strain, compatibility and equilibrium equations are rewritten in that frame, and a general solution for the isotropic case, is obtained in a fairly straightforward fashion. Polynomial expansions of the displacement field which are suitable for finite elements modeling of isotropic and anisotropic planes and that retains rotational degrees of freedom, are also presented.

This study contains two parts. The first one is a theoretical study to find mathematical model and for determination shear stresses in a thick walled cylinder with two circular holes under torsion moments using plane theory of elasticity and complex potential method which are defined its constant coefficients by jointly solution of three groups of Algebria linearly infinitely equations. The second parts contains general program written by (Q.Basic) language which can be used for the solution of Algebra equations with any limit number of equations taken form each group. And determination shear stresses for number of points located on the same axis .

We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory. The complexity of physical phenomena appears as encoded in so called material laws. The usefulness of the structural perspective developed is illustrated by showing that many initial boundary value problems in the theory of elastic solids share the same type of solution theory. Moreover, interconnections of the respective models are discussed via a previously introduced mother/descendant mechanism.

A method of reciprocal-space mapping followed by Rietveld-type refinement of the maps has been developed and tested for strongly textured thin films with fibre texture. The method is particularly useful for simultaneous analysis of stress and texture, especially in non-cubic materials. It could also be used for the analysis of other parameters, like film thickness, microstrain and domain size. Both of the extreme elastic models (Voigt/Reuss) have been adopted for the case of the fibre texture often present in thin films. The method allows estimation of residual stresses even for very strong [001] texture in hexagonal materials with angular half-widths of a few degrees. The procedure of twodimensional reciprocal-space map fitting is described in detail; simulations are presented and discussed. The method can be applied to multiphase materials, as long as the assumptions of the elastic models (Voigt/Reuss) are valid in the multiphase film. research papers 488 Daniel Simek et al. Reciprocal-space mapping

A problem about the stress-strain state of an elastic wedge with an arbitrary angel of opening, when on its bisector there is a system of a finite number collinear cracks, is studied by the means of the theory of elasticity. The anti-symmetric mixed boundary conditions given on both wedge-faces, together with the forces applied to the cracks&#x27; surfaces are provoking the anti-plane deformation of the wedge. The displacement components are given for the same group of nonintersecting intervals on each wedge-face and the stress components are given on the rest of the faces. The problem is formulated as a known mixed boundary problem of the theory of harmonic functions for a half-wedge because of the wedge symmetry relative to its bisector. The solution of this mixed boundary problem is derived in the closed form by using the Mellin integral transformation in combination with the methods of singular integral equations. Based on this the density of displacements&#x27; dislocations on ...

Classical continuum elastic calculations show that the acoustic vibrational modes of an embedded nanoparticle can be lightly damped even when the longitudinal plane wave acoustic impedances Zo = ρvL of the nanoparticle and the matrix are the same. It is not necessary for the matrix to be less dense or softer than the nanoparticle in order to have long lived vibrational modes. A corrected formula for acoustic impedance is provided for the case of longitudinal spherical waves. Continuum boundary conditions do not always accurately reflect the microscropic nature of the interface between nanoparticle and matrix, and a multi-layer model of the interface reveals the possibility of additional reduction of mode damping. 43.20.Ks,62.25.+g A classical vibrational mode weakly coupled to a cold environment will have an amplitude exponentially decaying with time. The vibrational frequency of the mode may be slightly shifted from what it would have been if the environmental coupling is reduced to zero. When coupling is stronger, the mode loses its individual identity, and it becomes necessary instead to consider normal modes of the entire system.

The paper analyses the stability of a rectangular plate which is elastically buckled along longitudinal edges and pressed by equally distributed forces. A general case is analyzed – different stiffness elastic clamping and then special simpler cases are considered. Energy method is used in order to determine critical stress. Deflection function is introduced in a convenient way so that it reflects the actual state of the plate deformation in the best manner. In this way, critical stress is determined in analytic form suitable for analysis. With help of the equation it is easy to conclude how certain parameters influence the value of critical stress. The paper indicates how the obtained solution could be utilized for determining local buckling critical stress in considerably more complex systems – pressed thin‐walled beams of an arbitrary length.

The theory of elastic contact between two spherical bodies according to Hertz, Mindlin and others is used as a basis for an extension to include the contribution of the viscous effects to the total stress for viscoelastic spheres subjected to twisting moments. Expressions relating twisting moment to the radius of a ‘stick’ region of the contact surface and the radius of the ‘stick’ region to twist angle are derived. Two term power series truncations of the relations are then used to derive approximate expressions for torsional stiffness of the bodies. Validation of the model was by experiments utilising a rheometer device. Applications for the model in post-harvest agriculture include extraction of material surface properties for use in discrete element modelling of mechanical interactions of fruits and other spheroidal produce during machine handling.

An exact solution of three-dimensional theory of elasticity in an infinite elastic rock medium around magma reservoirs of various aspect ratios and the criteria of rock fracturing provide an important tool for the interpretation of the origin and nature of distribution of many ore-bearing fracture systems around magmatic bodies in the world. Theoretical analyses suggest the conditions and regions of formation of (1) dominant radial fractures, (2) dominant concentric fractures or combination of radial and concentric fractures, and (3) central subsidence (or graben) in domal uplift related to intrusions of magma bodies of various shapes and sizes. The relations of these fracture systems to the deposition of ore-bearing fluids are discussed in the light of excess magma and hydrothermal fluid pressures. Caldera subsidence and dominant concentric fractures, such as in the San Juan Mountains mining district in Colorado, are, according to our analyses and interpretations, related to intrusion(s) of vertically elongated magma cupolas and high magma and hydrothermal fluid pressures. This differs from Anderson's original contention (1936) that caldera subsidence and outward-dipping ring fractures are due to lower relative magma pressure than lithostatic pressure. Wedging action of magma immediately above a vertically elongated prolate magma body is suggested as a cause of central graben bounded by funnel-shaped normal faults over wide domal uplift. Displacement along normal faults causes underground subsidence above magma cupolas. Extension of such faults close to the surface and displacement along them can produce surface caldera subsidence. Vein-type deposits commonly associated with funnel-shaped felsic (rhyolitic) bodies may be related to near-surface caldera subsidence. The origin of arcuate zones of breccia pipes such as in Silverton cauldron, Colorado, appears to be related to axial symmetric stress condition(s) due to vertically elongated magma body at depth. Estimation of stress distribution around spheroidal and prolate magma bodies suggests development of possible zonal distribution of fractures from the periphery of the magma outward in the order of (1) continuous tension fracture zone, (2) brittle fault zone, (3) ductile fault zone followed by no fault zone. Brittle fault and continuous tension fracture zones provide excellent sites for ore mineral localization. The zonal distribution of fractures also suggests. in a general way, mode of origin and possible extent of distribution of various fracture types, hydrothermal zoning, and their alterations.

The paper shows how a generalization of the elasticity theory to four dimensions and to spacetime allows for a consistent description of the homogeneous and isotropic universe, including the accelerated expansion. The analogy is manifested by the inclusion in the traditional Lagrangian of general relativity of an additional term accounting for the strain induced in the manifold (i.e. in space-time) by the curvature, be it induced by the presence of a texture defect or by a matter/energy distribution. The additional term is sufficient to account for various observed features of the universe and to give a simple interpretation for the so called dark energy. Then, we show how the same approach can be adopted back in three dimensions to obtain the equilibrium configuration of a given solid subject to strain induced by defects or applied forces. Finally, it is shown how concepts coming from the familiar elasticity theory can inspire new approaches to cosmology and in return how methods appropriated to General Relativity can be applied back to classical problems of elastic deformations in three dimensions.

In the monograph we investigate three{dimensional interface crack problems for metallic-piezoelectric composite bodies with regard to thermal effects. We give a mathematical formulation of the physical problems when the metallic and piezoelectric bodies are bonded along some proper parts of their boundaries where interface cracks occur. By the potential method the interface crack problems are reduced to equivalent strongly elliptic systems of pseudodifferential equations on manifolds with boundary. We study the solvability of these systems in appropriate function spaces and prove uniqueness and existence theorems for the original interface crack problems. We analyse the regularity properties of the corresponding thermo-mechanical and electric fields near the crack edges and near the curves where the different boundary conditions collide. In particular, we characterize the stress singularity exponents and show that they can be explicitly calculated with the help of the principal homo...

We discuss the concept of cloaking for elastic impedance tomography, in which, we seek information on the elasticity tensor of an elastic medium from the knowledge of measurements on its boundary. We derive some theoretical results illustrated by some numerical simulations.

The scope of the present article is the study of the fracture initiation from initial spherical flaws in incompressible propellant materials. Using a constitutive law, derived from the neo-Hookean strain energy function, one may deduce the relation between the radial pressure applied at infinity and the corresponding extension ratio, at the surface of the cavity. An equation which defines the critical extension ratio for a given value of a parameter, κ, was derived and the corresponding critical load, pc, is related to the critical cavity extension ratio. Plots for the extension ratio and normalized pressure were provided solving the appropriate equation derived from the analysis. The failure of the cavity is associated with the incremental energy transfer at a constant critical stress between the applied work, internal strain energy and surface energy. The influence of the surface energy on the radial deformation of an initially intact sphere is presented and it turns out that a characteristic cavitation phenomenon occurs instead of the bifurcation. A mathematical formula is derived for the pressure, per unit current area, which is valid for a general form of incompressible elastic strain energy function. Using the derived formula one may deduce the relation between the radial pressure and the corresponding extension ratio at the surface of the cavity. In an incompressible material of given modulus and surface energy it is proved that a critical field stress exists at which a small internal cavity may grow in size where the significant quantities are the radial and tangential strains.