Darboux classification of cauchy elastic bodies and some of its consequences

The Darboux theorem on the classification and canonical forms of first degree differential forms is used to obtain a classification and canonical representation of Cauchy elastic solids. If the stress is assumed to be determined uniquely by the Green deformation tensor C, the double Piola-Kirchhoff stress tensor is shown to belong to one of six possible classes, each of which possesses a unique representation in terms of linear combinations of gradients with respect to C of scalar-valued functions with scalar-valued coefficients. There are three distinct classes for isotropic materials, each of which has a unique representation. It is shown that these three classes are distinguishable in terms of the existence or nonexistence of cycles with nonzero total work and by the accessibility or inaccessibility of any deformation state from any neighboring deformation state by a path of zero work.     

Thermal-diffusion instability of the frictional contact of elastic bodies

We investigate the thermal-diffusion instability of the frictional contact of two half spaces with heat release. It is assumed that one of these spaces is an elastic heat-conducting binary solid solution, while the other is rigid and does not transfer heat or mass. By investigating the behavior of small surface perturbations of the concentration of the dissolved component, we found the critical sliding velocity beyond which perturbations exhibit a tendency to increase. It is shown that thermal diffusion is responsible for the instability of the contact, while diffusion induced by concentration gradients inhibits this process.Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 3, pp. 130–132, May – June, 1994.     

Refined solutions for the elastic response of involute bodies

A refined solution technique is presented for the involute model recently developed.^1,2 This consists of developing the proper relations, in particular, the end conditions, for a body composed of N sub-regions, associated with the finite difference solution scheme. An involute conical frustum problem treated earlier with one sub-region is solved by use of a two-sub-region model. Correlation with a fine network PATCHES-III solution is studied, with particular emphasis on predicted stress discontinuities at the (mathematical) interface of the two sub-regions. Observations regarding the general utility of the new model are also given.     

Thermodynamic influences on shock waves in inhomogeneous elastic bodies

Summary In this paper, the differential equation governing the behavior of shock waves propagating in inhomogeneous thermoelastic bodies is derived. The implications of this equation on the local behavior of the amplitudes are examined. It is shown that the behavior depends on the relative magnitudes of a number , called the critical jump in strain gradient, and the jump in strain gradient across the shocks.

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Thermodynamische Einflüsse auf Stoßwellen in inhomogenen elastischen Körpern

Zusammenfassung In dieser Arbeit wird die das Verhalten von in inhomogenen thermoelastischen Körpern sich ausbreitenden Stoßwellen charakterisierende Differentialgleichung abgeleitet. Die Folgerungen auf das lokale Verhalten der Amplituden wird untersucht. Es wird gezeigt, daß das Verhalten von der relativen Größe einer Zahl , genannt kritischer Sprung des Verzerrungsgradienten, und dem Sprung im Verzerrungsgradienten über den Stoß abhängt.

     

On the solution for the elastic response of involute bodies

A recent model to treat the elastic response of bodies of revolution possessing axisymmetric material structure, but otherwise arbitrary heterogeneous anisotropic elastic behavior, is examined in detail. Of particular interest is the derivation of the end conditions appropriate to the particular numerical solution technique employed (finite difference) because of the difference between the numbers of field equations and boundary conditions. It is felt that the procedure adopted may have much wider application than the present usage. The response of involute bodies is featured in this work as the illustrative problem of a frustrum of an involute cone under a constant temperature change is considered. A thorough comparison is made between the predictions given by the present model and those from the PATCHES-III finite element code, with generally favorable results.     

Moving punch on a highly orthotropic elastic layer

Summary An analytical solution was given to the problem of a long rigid punch moving rapidly on a strip of a highly orthotropic material. This elastic layer rests in turn on a rigid frictionless foundation. Obviously, the problem presents a lot of interest from the view-point both of geophysical and mechanical sciences. The solution was effected by means of integral transforms and the Wiener-Hopf technique. Asymptotic results were obtained valid near the edge of the moving punch.With 3 Figures     

On the elastic stability of thick columns

Summary Stability of a rectangular strip of isotropic highly elastic incompressible material is analyzed using the theory ofGreen, Rivlin andShield. The strip is subject to a uniaxial compression in planes of plane strain. Both the asymmetric and symmetric forms of buckling are studied and the results found in full agreement with the corresponding results obtained from the theory ofBiot.

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Zur elastischen Stabilität dicker Säulen

Zusammenfassung Die Stabilität eines rechteckigen Streifens aus isotropem, elastischem inkompressiblem Material wird mittels der Methode vonGreen, Rivlin undShield untersucht. Der Streifen steht unter einachsigem Druck in Ebenen des angenommenen ebenen Verzerrungszustandes. Sowohl die asymmetrischen als auch die symmetrischen Beulformen werden studiert. Die Ergebnisse befinden sich in voller Übereinstimmung mit den entsprechenden Resultaten der Theorie vonBiot.

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With 3 Figures

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Research supported by a grant of the National Science Foundation.     

A study on the linear elastic stability of Mindlin plates

The debate on the performance of Lagrangian and Serendipity elements in Mindlin's plate theory has been going on for quite some time. Limited published results for static and vibration analysis based on exact integration demonstrated a drastic deterioration in accuracy as the thickness of the plate decreases, and reduced/selective integration schemes have been suggested to improve their performance. Appreciable improvement for Lagrangian elements has been recorded, but it is only marginal for the Serendipity elements. On the strength of such observations one would then be tempted to rule out the exact integration schemes and Serendipity elements. In this paper, the above problem is reviewed for stability analysis of plates. Two elements are chosen from each family, one representing the higher order and the other the lower order element. Contrary to published results, all elements can attain very accurate solutions independent of the integration schemes for a sufficiently restrained plate, although in general the Serendipity elements will require a more refined mesh than the Lagrangian ones. However, for loosely restrained plates, the solution failed when integration is performed by reduced/selective schemes. The failure marks the limitation of the reduced/selective schemes which have somehow introduced spurious modes into the system, but on the other hand it is ironical that these spurious modes in fact contribute to the improvement of performance of the restrained cases. Therefore, one can equally improve the Serendipity elements by a selective scheme which can introduce additional zero modes. A scheme based on (5 × 5) integration points for flexural stiffness and (3 × 3) integration points for shear stiffness improved the 17SE (Serendipity elements) remarkably. However, because of the lack of bounds in most cases, the use of reduced/selected schemes is still not recommended. Finally, this paper also proposed an approximate formulation of the geometric stiffness matrices to replace the full formulation. Such an approximate formulation reduces the number of variables considerably in the eigenvalue search and can still give reasonably accurate results for thin plates with a/t > 15.     

On complex potentials in the theory of microstretch elastic bodies

The paper is concerned with the plane strain problem in the equilibrium theory of microstretch elastic solids. We show that the complex variable technique of the classical theory of elasticity can be extended to the theory of microstretch elastic bodies. The method is used to study the effect of the stress concentration around a circular hole.     

Topological derivatives for fundamental frequencies of elastic bodies

In this article a new method for topological optimization of fundamental frequencies of elastic bodies, which could be considered as an improvement on the bubble method, is introduced. The method is based on generalized topological derivatives. For a body with different types of inclusion the vector genus is introduced. The dimension of the genus is the number of different elastic properties of the inclusions being introduced. The disturbances of stress and strain fields in an elastic matrix due to a newly inserted elastic inhomogeneity are given explicitly in terms of the stresses and strains in the initial body. The iterative positioning of inclusions is carried out by determination of the preferable position of the new inhomogeneity at the extreme points of the characteristic function. The characteristic function was derived using Eshelby's method. The expressions for optimal ratios of the semi-axes of the ellipse and angular orientation of newly inserted infinitesimally small inclusions of elliptical form are derived in closed analytical form.     

Dynamic interaction of parallel cracks in elastic bodies

We study time dependences of the stress intensity factors in an infinite isotropic elastic body with parallel cracks whose faces are subjected to impact loads. The original problem is reduced to the solution of a system of boundary integral equations with a parameter corresponding to the wave number in the decomposition of the dynamic process into the set of monochromatic components. The numerical reconstruction of the dependences of unknown quantities on time is carried out for two circular cracks loaded by tensile forces in the form of the Heaviside function. The presence of stress intensity factors of all three types is a distinctive feature of the considered problem. Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 94–102, May–June, 1998.     

Some questions of stability of the plane form of deformation of filaments of finite rigidity beyond the elastic limits

A method of calculating the stabilty of the plane form of bending of elastoplastic filaments of finite rigidity is presented on the basis of the theory of plastic flow. It is shown that this problem reduces to relationships similar to Kirchhoff's equations for an elastic case of working. Dependences containing equations of beam theory as a particular case are obtained for determining the reduced rigidity characteristics of the cross section of a filament with plastic regions. The article investigates the effect of size of the plastic regions on the interface between the plastic-deformation and load relief zones and also on the cross section of the filament. Graphically represented results of calculations are analyzed.Translated from Problemy Prochnosti, No. 10, pp. 37–44, October, 1992.     

Some results on finite amplitude elastic waves

Summary 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.

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Zusammenfassung Es werden Spezialfälle von Wellenbewegungen mit endlicher Amplitude betrachtet, für die keine kinematischen Nichtlinearitäten in den Bewegungsgleichungen des elastischen Festkörpers auftreten. Es kommen also nur Material-Nichtlinearitäten vor. Für gewisse Klassen von Materialien können die Lösungen aus den entsprechenden der linearen Theorie gewonnen werden. Hierher gehören horizontal polarisierte Scherwellen und Lovesche Wellen in geschichteten oder inhomogenen Medien. Es wird gezeigt, daß sich harmonische, zirkularpolarisierte Transversalwellen in jedem kompressiblen oder inkompressiblen elastischen Material fortpflanzen. Einige Effekte homogener Vorspannung werden ebenfalls untersucht.

     

A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies

Recently, there has been interest in the study of a new class of constitutive relation, wherein thelinearized strain tensor is assumed to be a function of the stresses. In this communication, some boundaryvalue problems are solved using the finite element method and the solid material being described by such aconstitutive relation, where the stresses can be arbitrarily ‘large’, but strains remain small. Three problems areanalyzed, namely the traction of a plate with hyperbolic boundaries, a plate with a point load, and the tractionof a plate with an elliptic hole. The results for the stresses and strains are compared with the predictions thatare obtained by using the constitutive equation of the classical linearized theory of elasticity.     

On the nonlinear elastic response of bodies in the small strain range

The classical linearized approximation to describe the elastic response of solids is the most widely used model in solid mechanics. This approximate model is arrived at by assuming that the norm of the displacement gradient is sufficiently small so that one can neglect the square of the norm in terms of the norm. Recent experimental results on Titanium and Gum metal alloys, among other alloys, indicate with unmistakable clarity a nonlinear relationship between the strain and the stress in the range of strain wherein one would have to use the classical linearized theory of elasticity, namely wherein the square of the norm of the strain can be ignored with regard to the value of the strain, leading to a dilemma concerning the modeling of the response, as the classical nonlinear Cauchy elastic model would collapse to the linearized elastic model in this range. A novel and important generalization of the theory of elastic materials has been suggested by Rajagopal in Appl Math 48: 279–319, 2003 and Zeit Angew Math Phys 58: 309–317, 2007 that allows for an approximation wherein the linearized strain can be a nonlinear function of the stress. In this paper, we show how this new theory can be used to describe the new experiments on Titanium and Gum metal alloys and also clarify several issues concerning the domain of application of the classical linearized theory.     

Measurement of parameters of the high-speed motion of bodies along elastic guides

A new principle is proposed for measuring parameters of the high-speed motion of bodies along elastic guides. The principle is based on ultrasonic probing of the guides. Translated from Izmeritel'naya Tekhnika, No. 6, pp. 44–46, June, 1999.