Showing papers in "Acta Mechanica in 1967"
TL;DR: In this article, the authors considered some special types of finite amplitude wave motions, for which kinematical non-linearities do not arise in the equations of motion of an elastic solid.
Abstract: In this paper some special types of finite amplitude wave motions are considered, for which kinematical non-linearities do not arise in the equations of motion of an elastic solid. Consequently, only constitutive non-linearities occur and, for special classes of materials, solutions may be read off from corresponding solutions in the linear theory. These include SH-waves1 and Love waves in layered or inhomogeneous media. Finite amplitude transverse circularly-polarized harmonic progressive waves are shown to propagate in any compressible or incompressible isotropic elastic material. Some effects of homogeneous pre-stress are also investigated.
69 citations
TL;DR: In this article, explicit forms of the constitutive relations, the dissipation function and related thermodynamical restrictions are derived for infinitesimal non-isothermal deformation of linear viscoelastic solids.
Abstract: With the use of a suitable representation for the free energy, explicit forms of the constitutive relations, the dissipation function and related thermodynamical restrictions are derived for infinitesimal non-isothermal deformation of linear viscoelastic solids. Some remarks on further restrictions, consistent with the dissipation inequality, are included for isotropic materials.
61 citations
TL;DR: In this paper, the problem of plane strain compression between rigid, parallel plates of a long slab of an ideal soil or granular medium was considered and closed form solutions for the stress and velocity fields were presented and an expression for the load carrying capacity of the soil was derived.
Abstract: The problem considered here is that of the plane strain compression between rigid, parallel plates of a long slab of an ideal soil or granular medium. Two situations are examined (a) a cohesive soil compressed between perfectly rough plates, (b) the compression of a cohesionless soil with sliding friction at the plates. Closed form solutions for the stress and velocity fields are presented and an expression for the load carrying capacity of the soil is derived for the case when the weight of the soil is negligible. The solutions are applicable to a particular foundation problem. The effect of the weight of the material will be considered in future work.
40 citations
TL;DR: In this article, the results of an earlier formulation of the problem of elastic plates on an elastic foundation are restated in more compact and explicit form, and the earlier results are supplemented in an important way by giving transition conditions for the interior of the foundation layer along the cylindrical surface.
Abstract: The results of an earlier formulation of the problem of elastic plates on an elastic foundation are restated in more compact and explicit form. The earlier results are supplemented in an important way by giving transition conditions for the interior of the foundation layer along the cylindrical surface which separates loaded and unloaded portions of the foundation.
29 citations
TL;DR: In this paper, it was shown that the classic stability theory of thin plates and shells should be reconsidered taking into consideration the angles of rotation, and the results of the present paper confirm that when q = p the solid can become unstable which contradicts the findings of other investigators.
Abstract: The paper starts with a critical survey of recent investigations of related problems. The title problem shown in Fig. 2 is then solved whenq is constant directional and whenq is hydrostatic, using the complete formulation given byNovozhilov with the only assumption being that the elongations and shears are small compared to unity. Since the angles of rotation were retained in the formulation, and the obtained expressions did reduce for thin plates to the correspondingEuler-type expressions, it is concluded thatLegenya's suggestion, that the classic stability theory of plates and shells should be reconsidered taking into consideration the angles of rotation, is based on erroneous results. The results of the present paper confirm that whenq=p the solid can become unstable which contradicts the findings of other investigators. The paper concludes with a comparison of “buckling pressure versus slenderness ratio” curves which are based on results of a number of investigators. These graphs demonstrate that the classical theory of buckling of thin plates (and beams) yields accurate results up toh/L=0.15, thus verifying the validity of a usual assumption of structural mechanics.
28 citations
TL;DR: In this paper, the influence of thermomechanical coupling on the magnitudes of discontinuities at the wavefronts of dilatational waves is studied, and it is shown that material damping is predominant except at extremely small distances from the center which fall outside the realm of continuum theory.
Abstract: The method of the characteristics is used to study the influence of thermomechanical coupling on the magnitudes of discontinuities at the wavefronts of dilatational waves. A set of unified equations is employed, which is applicable to plane, cylindrical and spherical waves. The thermal conductivity is taken as a function of the space coordinate. For cylindrical and spherical symmetry the discontinuity at the wavefront of a diverging wave is subjected to both material damping and decay due to geometry. The sharp wavefront is attenuated over a very short distance. For converging waves the material damping is counteracted by an increase in magnitude due to geometry. It is shown that material damping is predominant, except at extremely small distances from the center which fall outside the realm of continuum theory.
22 citations
TL;DR: In this paper, the authors analyzed the stresses in micropolar elasticity around a circular hole in an infinite plate, due to prescribed tractions in the plane of the plate, acting on the circular boundary.
Abstract: This paper analyses the stresses in micropolar elasticity around a circular hole in an infinite plate, due to prescribed tractions in the plane of the plate, acting on the circular boundary. It is found that the stresses and couple stresses are controlled byPoisson's ratio and two new parameters which depend on the material constants of the micropolar medium.
22 citations
TL;DR: The method of point matching was used in this paper to solve three problems for the bending of a plate having circular holes, i.e., deflection and bending moment curves along the hole and along the outer edges for various ratios of hole diameter to size of plate.
Abstract: The method of point matching is used to solve three problems for the bending of a plate having circular holes. The first two problems consist of a uniformly loaded square plate either simply supported or clamped along the outer boundary. Free edge boundary conditions are satisfied exactly along the internal hole and the conditions along the outside contour are satisfied by point matching in the least squares sense. Deflection and bending moment curves along the hole and along the outer edges are presented for various ratios of hole diameter to size of plate. Results for the case when the outer edges are simply supported can be compared with the less extensive results ofDedic. The last problem solved is that of an infinite plate having equally spaced circular holes. The plate is loaded by its own weight and supported at points equidistant from the hole centers. Three different approaches to the problem are used, all satisfying the boundary conditions by point matching. Results for deflections and bending moments for various hole diameters are presented.
20 citations
TL;DR: In this article, a coupled energy equation is used to calculate the transient and steady-state temperature distributions in a thin-walled tube subjected to torsional oscillations, and the heating is determined solely by the materials mechanical behavior and by consistency with thermodynamics.
Abstract: The temperature distributions in a viscoelastic solid which result from mechanical deformations are investigated. The solid is assumed to be isotropic and linear. A coupled energy equation is used to calculate the transient and steady-state temperature distributions in a thin-walled tube subjected to torsional oscillations. The heating is determined solely by the materials mechanical behavior and by consistency with thermodynamics.
19 citations
TL;DR: In this paper, a linearized theory is obtained from the nonlinear theory of elasticity to investigate waves and vibrations in an elastic solid under initial stress, and it is found that uniform shear stress is able to induce coupling of dilatational and equivoluminal waves and to alter the speed of propagation.
Abstract: FollowingNovozhilov andBolotin, a linearized theory is obtained from the nonlinear theory of elasticity to investigate waves and vibrations in an elastic solid under initial stress. Wave propagation in an infinite medium is studied for three cases: (1) hydrostatic compression or tension, (2) uniaxial extension or compression, (3) uniform shear stress (for the case of plane strain). It is observed that hydrostatic pressure and uniaxial extension affect the rate of propagation. Further, uniform shear stress is found to induce coupling of dilatational and equivoluminal waves and to alter the speed of propagation. Results are discussed and compared with other theories.
19 citations
TL;DR: In this article, the development of the field equations and constitutive equations for small motions superposed on large static deformation of a liquid-filled porous elastic solid, using the basic theory of interacting continua given in [3], is discussed.
Abstract: This paper is concerned with the development of the field equations and constitutive equations for small motions superposed on large static deformation of a liquid-filled porous elastic solid, using the basic theory of interacting continua given in [3]. Two examples of parallel flow in a medium which initially is subjected to a homogeneous deformation are discussed. Also a representation for the solutions of the equations of the infinitesimal theory (analogous to theBoussinesq-Papkovich representation) is deduced for a class of steady flow.
TL;DR: In this paper, the problem of local stability of structural systems subjected to conservative and non-conservative external forces is examined and it is shown that, contrary to the usual assumption and in contrast to the case of conservative loading, if linearized analysis is used for local stability investigation of nonconservative systems, then the existence of simple harmonic vibrations of constant amplitudes does not necessarily imply stability.
Abstract: The problem of local stability of structural systems subjected to conservative and nonconservative external forces is examined. It is shown that, contrary to the usual assumption and in contrast to the case of conservative loading, if linearized analysis is used for local stability investigation of nonconservative systems, then the existence of simple harmonic vibrations of constant amplitudes does not necessarily imply stability. In this case, the local stability of the equilibrium can only be decided upon by a nonlinear analysis. Moreover it is shown that, for such systems, the linearized analysis yields the true “critical” force only if all the dissipative forces are included directly in the analysis. Finally, certain thermodynamic implications of these observations are pointed out.
TL;DR: In this paper, the authors apply the Coleman-Noll approach to the thermodynamics of materials and deduce the essential feature of plasticity in a rational way, which is formally of the Drucker type.
Abstract: TheColeman-Noll approach to the thermodynamics of materials is applied to the case of plasticity. The essential constitutive property is that the dependent thermodynamic variables are functions of the deviators of both the stress and strain, but are otherwise independent of time in the same sense as elastic substances. The conclusion is then deduced, from the second law, that for such materials the loading and unloading stress-strain relations must be different; thus the essential feature of plasticity is deduced in a rational way. The second law is also shown to imply a material stability which is formally of theDrucker type.
TL;DR: In this paper, the authors show how das Verfahren vonGalerkin can be noch konvergiert, wenn man Ansatzfunktionen verwendet, die nicht alle Randbedingungen exakt erfullen, vorausgesetzt, das man das VERFahren dann durch die "Mitnahme" von gewissen Randtermen erweitert.
Abstract: Es wird gezeigt, das das Verfahren vonGalerkin auch dann noch konvergiert, wenn man Ansatzfunktionen verwendet, die nicht alle Randbedingungen exakt erfullen, vorausgesetzt, das man das Verfahren dann durch die “Mitnahme” von gewissen Randtermen erweitert.
TL;DR: In this paper, the problem of torsion of prismatical bars is reduced to an inhomogeneousFredholm integral equation for the warping function on the boundary of the cross-section.
Abstract: ZusammenfassungAlle bisher bekannten exakten wie auch Näherungsmethoden zur Lösung des Torsionsproblems hängen von der Gestalt des betreffenden Querschnittes ab. Von den vorhandenen speziellen Lösungen besitzt nur ein kleiner Teil praktische Bedeutung.Im folgenden wird eine Lösungsmethode angegeben, die für jede Art von Querschnitt Verwendung finden kann. Das Torsionsproblem wird mit Hilfe desCauchyschen Integralsatzes auf eine inhomogeneFredholmsche Integralgleichung für die Wölbfunktion am Querschnittsrand zurückgeführt und mittels desNyström-Verfahrens gelöst. Wenn die Wölbfunktion entlang der Querschnittsberandung bekannt ist, können Torsionsspannungen und Torsionsträgheitsmomente auf einfache Weise bestimmt werden.Die Methode wird auf einige in der technischen Praxis vorkommende Beispiele angewendet. Ihr größter Vorteil liegt in der Möglichkeit, Querschnitte mit einspringenden Ecken zu behandeln.SummaryAll known methods, exact or approximate, for the solution of the torsional problem depend on the particular shape of the cross-section. Of the special solutions available only a small fraction, is of practical value.In the following a method is presented which may be applied to any type of cross-section. With the aid of theCauchy integral theorem the problem of torsion of prismatical bars is reduced to an inhomogeneousFredholm integral equation for the warping function on the boundary of the cross-section. TheNyström method is used for the solution of the integral equation. Once the warping function is known on the boundary torsional stress and torsional stiffness may be determined in a simple manner.The method is applied to particular examples of practical interest. Its main advantage appears to be the possibility to treat cross-sections with re-entrant corners.
TL;DR: In this article, a square epoxy slab was bonded to a rigid plate on one of its faces, and a state of restrained shrinkage developed, and slices removed from the slab were used to determine the stress.
Abstract: A square epoxy slab was bonded to a rigid plate on one of its faces. In the process of “freezing” photoelastic effects a state of restrained shrinkage developed. Slices removed from the slab were used to determine the stress.
TL;DR: In this paper, the optimal design of reinforced slabs is analyzed assuming a rigid, perfectly-plastic model of both concrete and reinforcement, and two approximations of the nonlinear cost function are introduced.
Abstract: A problem of optimum design of reinforced slabs is analyzed assuming a rigid, perfectly-plastic model of both concrete and reinforcement. For a slab of constant thickness such static field is sought which minimizes the total amount of reinforcement under constant limit load; for a slab of varying thickness, the total cost of materials is assumed as a design criterion. Two approximations of the nonlinear cost function are introduced and the corresponding static and kinematic relations are discussed in detail. Several examples of circular and annular slabs under symmetric and non-symmetric loading are considered in order to illustrate the theory.
TL;DR: In this article, a class of nonlinear constitutive equations of rate-type are considered for viscoelastic-plastic continua and explicit restrictions are derived from thermodynamics, and some aspects of the associated linearized theory with isothermal deformations are also discussed.
Abstract: A class of nonlinear constitutive equations of rate-type are considered for viscoelastic-plastic continua and explicit restrictions are derived from thermodynamics. Some aspects of the associated linearized theory with isothermal deformations are also discussed.
TL;DR: In this paper, a rough loose-fitted rigid pin is pressed against one portion of the boundary of the hole, thus giving a specified displacement on the contact area while the rest of the perimeter is stress free.
Abstract: This paper is concerned with an infinite plate of homogeneous elastic material in a state of generalized plane stress and having a circular hole. A rough loose-fitted rigid pin is pressed against one portion of the boundary of the hole, thus giving a specified displacement on the contact area while the rest of the boundary is stress free. The problem has been solved after converting it into aHilbert's Problem. Values of stresses at different points of the boundary have been calculated.
TL;DR: Aus der Betrachtung der Grundgleichungen der Kontinuumsmechanik an materiellen Unstetigkeitsflachen folgt eine anschauliche Deutung der verallgemeinerten Dissipationsungleichung as mentioned in this paper.
Abstract: Aus der Betrachtung der Grundgleichungen der Kontinuumsmechanik an materiellen Unstetigkeitsflachen folgt eine anschauliche Deutung der verallgemeinerten Dissipationsungleichung.
TL;DR: In this article, a solution of the displacement equations for the case of fluid sources is given with the aid of the principle of reciprocity of displacements, and the singular solution is obtained with the action of the fluid sources.
Abstract: The present paper is based onBiot's theory of flow of fluids through a porous deformable medium. The presence of a distribution of sources of fluids in the medium gives rise to an additional term in the generalizedDarcy's law. The present note is concerned with solutions of the displacement equations ofBiot's theory for the case of fluid sources. The regular solution of these equations is given in terms of special functions. The singular solution — connected with the action of fluid sources — is obtained with the aid of the principle of reciprocity of displacements.
TL;DR: In this paper, the authors extend the class of functions that define displacement fields that are complete to include all displacement fields defined in terms of the Lame scalar and vector potentials, requiring only that theLame potentials satisfy the two wave equations resulting from the displacement equations of motion.
Abstract: An aid to the solution of the displacement equations of motion in the linear theory of homogeneous and isotropic elastic solids is the decomposition of the displacement field into a rotational and irrotational part, the so-calledLame scalar and vector potentials. Previous works have required the vector potential to be divergence free in order to insure completeness of this representation of the displacement field. The purpose of the present paper is to broaden the class of functions that define displacement fields that are complete to include all displacement fields defined in terms of theLame potentials, requiring only that theLame potentials satisfy the two wave equations resulting from the displacement equations of motion.
TL;DR: In this paper, a study has been made of a discontinuous boundary-value problem, where a circular disc made of homogeneous isotropic elastic matcrial is in a state of generalized plane stress.
Abstract: In this paper a study has been made of a discontinuous boundary-value problem. A circular disc made of homogeneous isotropic elastic matcrial is in a state of generalized plane stress. The boundary of the disc is divided into two parts. Over one part the stresses are zero; over the other the shear stress is zero and the normal displacement is specified. The problem has been solved after converting it into aPrandtl-type integral equation.
TL;DR: In this article, a superposition principle was used to obtain exact solutions to a plane, a cylindrical, and a two-dimensional wave problem in a viscoelastic material with constant Poisson's ratio.
Abstract: A superposition principle obtained recently by the author, reduces the viscoelastic wave problem to a static elastic problem, an elastic eigenvalue problem and an integrodifferential equation of theVolterra type involving time only. This principle is utilized to obtain exact solutions to a plane, a cylindrical, and a two-dimensional wave problem in a viscoelastic material with constantPoisson's ratio.
TL;DR: In this paper, a new direct numerical method for finding optimal values of functionals and associated velocity fields is described, which replaces a continuous problem by a discrete problem, which is then solved routinely on a high-speed digital computer.
Abstract: Interpreted in terms of the Limit Design Theorems, a problem in plastic analysis of a structure is equivalent to finding the minimum of a functional over a set of appropriate velocity field functions. In this paper we describe a new direct numerical method for finding optimal values of such functionals and associated velocity fields. The technique, which makes use of the methods available for solving problems in mathematical programming, consists of replacing a continuous problem by a discrete problem, which is then solved routinely on a high-speed digital computer.
TL;DR: Within the framework of the linearized theory of couple-stresses for perfectly elastic, non-homogeneous, anisotropic materials, a general reciprocal theorem is derived and applied in the proof of other general theorems in both elastostatics and elastodynamics.
Abstract: Within the framework of the linearized theory of couple-stresses for perfectly elastic, non-homogeneous, anisotropic materials, a general reciprocal theorem is derived and applied in the proof of other general theorems in both elastostatics and elastodynamics. In particular, a general work theorem is derived; for small periodic vibrations it is shown that the normal mode functions are orthogonal; and these results are combined to prove that if the strain energy function is positive definite, then the normal mode frequencies are real and non-vanishing. In addition,Castigliano's theorem and several other theorems that are common to structural analysis are shown to follow in a rigorous and natural way from the theory of couple-stresses.
TL;DR: In this article, the authors discuss the Verhalten eines zwischen zwei parallelen Holmen aufgespannten biegeschlaffen Tuches unter dem Einflus der Stromungskrafte.
Abstract: Das Verhalten eines zwischen zwei parallelen Holmen aufgespannten biegeschlaffen Tuches unter dem Einflus der Stromungskrafte wird fur die Falle der Anstromung mit Uberschall- und mit Hyperschallgeschwindigkeit sowie mit schallnaher Uberschallgeschwindigkeit angegeben. Aus den allgemeinen Beziehungen fur das Kraftegleichgewicht am Tuch und aus den Gleichungen fur die Druckkrafte infolge der Stromung erhalt man unter der Voraussetzung kleiner Anstellwinkel und schwacher Profilwolbungen in allen Fallen einfache gewohnliche Differentialgleichungen erster Ordnung fur die Profilneigung. Bei Uberschallgeschwindigkeit ergibt sich die Druckdifferenz am Tuch aus der Umstromung schlanker Profile unter der Verwendung linearisierter Gleichungen. Im Hyperschallbereich wird die sogenannte tangent-wedge-Methode benutzt. Im schallnahen Gebiet erhalt man die benotigten Druckkoeffizienten aus Entwicklungen derPrandtl-Meyer-Expansion und der Stospolarengleichung fur Schallnahe. Das Verhalten der Profilkontur in Abhangigkeit von den Parametern Anstellwinkel, Tuchlange und Anstrommachzahl wird diskutiert, ebenso das des Auftriebs und des Widerstandes. Es ergeben sich sowohl in den Konturformen als auch in deren Stabilitatsverhalten Unterschiede beim Uberschall- und beim Hyperschallgebiet gegenuber dem Unterschall-und dem Schallnahebereich. Der Einflus des Tucheigengewichtes wird angegeben. Die aus den vorgenommenen Vernachlassigungen herruhrenden Grenzen der Anwendungsmoglichkeit der vorliegenden Naherungen werden abgesteckt.
TL;DR: In this paper, the electromechanical equations of displacement were solved to generate the frequency equation for the pure radial modes of a barium-titanate shell and numerical values were tabulated for the first three frequencies for a broad range of shell thicknesses.
Abstract: A thick-shell hollow piezoceramic sphere electroded completely on both inside and outside surfaces and radially polarized can be used as a source and receiver of underwater sound. Restricting the analysis to moderately thick shells the electromechanical equations of displacement are solved to generate the frequency equation for the pure radial modes. The frequency equation is specialized to reflect the behavior of a barium-titanate shell and numerical values are tabulated for the first three frequencies for a broad range of shell thicknesses.
TL;DR: The Verzweigungspunkt im nichtlinearen Spannungs problem kann exakt aus den linearisierten Beziehungen erhalten werden, wenn der geschlossene Kreisring unter konstantem Ausendruck steht, da hier im stabilen Falle die Form der Stabachse ahnlich dem Ausgangszustand ist as mentioned in this paper.
Abstract: Der Verzweigungspunkt im nichtlinearen Spannungsproblem kann exakt aus den linearisierten Beziehungen erhalten werden, wenn der geschlossene Kreisring unter konstantem Ausendruck steht, da hier im stabilen Falle die Form der Stabachse ahnlich dem Ausgangszustand ist. Die Untersuchungen werden fur vier Belastungen durchgefuhrt, die sich durch ihr Verhalten wahrend des Knickvorganges unterscheiden.