Showing papers in "Acta Mechanica in 1972"

A multi-continuum theory for composite elastic materials

Abstract: A continuum theory for composite materials is presented in which the composite constituents are modeled by superimposed continua which undergo thermal and mechanical interactions. Kinematical notions, field equations and a constitutive theory are developed, including consequences of material frame indifference and material symmetry which restrict the form of the response functions. The final result is a set of linearized equations for thermal and mechanical processes in laminated or fiber reinforced composite elastic materials.

An experimental investigation of yield surfaces at elevated temperatures

Abstract: The purpose of this paper is to present new methods and procedures used in the experimental determination of yield surfaces at room and at elevated temperatures These procedures permit one to obtain a striking consistency in the results and to derive unambiguously laws of general validity Experiments are presented in which thin-walled tubes of pure aluminum are loaded in combined tension and torsion, in the plastic range, at room temperature and at elevated temperatures to 325°F Yield surfaces in stress space are obtained at several temperatures for the virgin material and for the material prestrained in torsion It is shown that no cross effect exists at all tested temperatures and levels of prestraining It is also shown that the yield surfaces do not pass through the prestressing point and that the yield surfaces do not enclose the origin even at very small values of prestressing The paper ends with the introduction of a new method of determining the yield surface in stress-temperature space In this method the loading path is non-isothermal and the experimental results verify the previously isothermally determined yield surface

Dreidimensionale Störungen in der Grenzschicht an einer konkaven Wand

Abstract: In dieser Arbeit wird experimentell die Entwicklung von dreidimensionalen Storungen in der Grenzschicht an einer parallel angestromten konkaven Wand untersucht Durch photogrammetrische Auswertung von Stereoaufnahmen der mit Hilfe der Wasserstoffblaschenmethode sichtbar gemachten Stromung werden Geschwindigkeitsverteilungen in der instabilen Grenzschicht sowie die Anfachung kleiner erzwungener Anfangsstorungen gemessen und mit der linearen Theorie verglichen Im nichtlinearen Bereich der Anfachung wird in Langszonen kleiner GeschwindigkeitU die Entstehung einer sekundaren Instabilitat beobachtet Es wird gezeigt, wie eine damit verbundene Schlangelbewegung schlieslich den laminar-turbulenten Umschlag herbeifuhrt

The Deformations and Stresses in Floating Ice Plates

Abstract: In the past, the analyses of floating ice plates subjected to static or dynamic loads were based on the theory of a thin homogeneous plate, although in actual floating ice plates Young's modulus may vary strongly with depth. Recently,A. Assur concluded, on the basis of a heuristic argument, that the solutions obtained for homogeneous plates may be used for floating ice plates, if a modified flexural rigidity is used. The purpose of the present paper is to study this question, by establishing a mathematically consistent formulation for the dynamic plate equation, utilizing Hamilton's Principle in conjunction with the three dimensional theory of elasticity. It is proven that for a variable Young's modulus and a constant Poisson's ratio the resulting formulations for plates and beams are the same as those for the corresponding homogeneous problems, if a modified flexural ridigity is used; thus confirmingAssur's conclusion. It is shown that the stress distribution is not linear and that the stress formula\(\sigma _{\max } = M{{z_0 } \mathord{\left/ {\vphantom {{z_0 } I}} \right. \kern- ulldelimiterspace} I}\) used by a number of investigators for the determination of the carrying capacity of a floating ice plate, as well as for the computation of failure stresses from tests on floating ice beams, is not applicable. Correct formulas are derived, corresponding stress distributions are presented and the consequences of the findings discussed.

Wave propagation in a micropolar elastic half-space

Abstract: Wave propagation in an infinite micropolar elastic half space and the reflection of plane longitudinal displacement waves from a fixed flat surface of a micropolar elastic half space are investigated. Reflection laws and amplitude ratios are presented for specific cases. New propagating and reflected waves are found in addition to the classical ones.

A Form of the Translational Dynamical Equations for Relative Motion in Systems of Many Non-Rigid Bodies*

Abstract: The translational dynamical equations for an arbitrary system of interconnected non-rigid bodies is written in terms of the relative translations between bodies, the barycentric vectors describing attachment points between bodies, and the deformational deflections of those points. The results are derived in terms of the characteristics of the system graph, in the spirit of the Roberson-Wittenburg formulation of the rotational dynamical equations.

Experimental Investigation of Corners in the Yield Surface

Abstract: Thin-walled tubular specimens of 1100 aluminum and OFHC copper were subjected to biaxial zig-zag loading paths through the application of simultaneous axial tension and internal pressure. The definite correlation between the change in direction of the applied stress increment and the resulting plastic strain increment indicated the existence of a corner or pointed vertex at the loading point. Direct yield surface determinations both before and after the zig-zag loading tests indicated smooth yield curves. The apparent disagreement was a result of effects associated with the unloading process prior to the yield surface determination. It is postulated that the loading surface (boundary between elastic and plastic deformation during loading) contains a corner at the loading point, and the yield surface (elastic-plastic boundary in the unloaded state) remains smooth. Hence, both the zig-zag loading results, which depend upon the shape of the loading surface, and the yield surface studies, which depend upon the shape of the yield surface, are explained.

Wave propagation in elastic laminates using a multi-continuum theory

Abstract: A general continuum approach to the dynamical behavior of elastic composites proposed in an earlier paper is specialized to infinitesimal deformations of elastic laminates. A simple procedure is proposed to evaluate the constitutive parameters appropriate to a particular choice of constituents, which is easily generalized to other types of material behavior and composite structure. The resulting theory is then used to investigate the propagation of plane harmonic waves to furnish a basis for comparison with other dynamical treatments of elastic laminates.

Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden

Abstract: Messungen des Anwachsens von Langswirbeln in zeitlich anwachsenden Grenzschichten an konkav gekrummten Wanden (Gortler-Taylor-Wirbel) ergaben drei deutlich getrennte Bereiche: Es traten zunachst Wirbel mit der Wellenlage λ≈0,9δ auf (δ=Grenzschichtdicke, λ=Hohe einer Zelle, die zwei gegensinnig drehende Wirbel enthalt). Je nach Grose der mit der Verdrangungsdicke δ1 der Grenzschicht gebildeten Reynolds-Zahl\(\frac{{\omega R_a \delta _1 }}{v}\) erschienen dann kurze Zeit spater Wirbel mit λ≈ 2,5δ, wenn\(\operatorname{Re} _{\delta _1 } \leqq 276\) war. Im Bereiche\(\operatorname{Re} _{\delta _1 } \geqq 312\) dagegen traten stattdessen bei den hier durchgefuhrten Versuchen immer Wirbel mit der Wellenlange λ≈6,5 δ auf. Bei\(\operatorname{Re} _{\delta _1 } \geqq 420\) werden die ersten Tollmien-Schlichting-Wellen mit der Wellenlange λ TS ≈6·δ angefacht. In ihren wandnahen Bereichen der Wellentaler konnten sich dann die oben genannten Langswirbel der Wellenlange λ≈6,5·δ ausbilden, die die zwei-in eine dreidimensionale Storung allseits gleicher Grosenordnung verwandeln konnen.

Finite buckling of fiber-reinforced columns

Abstract: Finite, plane buckling deformations of slabs or columns composed of initially straight, parallel fibers embedded in a weaker matrix are discussed. The fibers are treated as continuously distributed and inextensible, and the composite is treated as incompressible. It is found that the critical buckling load, per unit of length normal to the plane of deformation, isGH, the infinitesimal shear modulusG times the widthH of the column. Stability of buckled states is considered. Limits on the applicability of the present theory to real composites are discussed.

Zur analytischen Erfassung des allgemeinen Bauschingereffektes

Abstract: Unter dem allgemeinen Bauschingereffekt sollen gewisse Besonderheiten im Verlaufe der Fliesspannungen im Anschlus an eine beliebige Anderung des Flieszustandes verstanden werden. Es werden zwei Anteile des Bauschingereffektes unterschieden: ein verschwindender und ein bleibender Anteil. Bisherige analytische Ansatze fur eine kinematische Verfestigung erfassen nur den zweiten Anteil. Der erste Anteil wurde in einer fruheren Untersuchung des Verfassers dargestellt. Eine Ubereinstimmung mit Versuchsergebnissen last sich jedoch nur erreichen, wenn zwei weitere Effekte als vorhanden angenommen werden. Der erste besteht in einer teilweisen Relaxation der zum kinematischen Anteil gehorenden Spannungen, der zweite in einer isotropen Entfestigung, die sich in einer Erhohung des zweiten Anteils auswirkt. Die Berucksichtigung dieser beiden zusatzlichen Effekte durch geeignet erscheinende analytische Ansatze fuhrt zu einer guten quantitativen Ubereinstimmung mit den betrachteten Versuchsergebnissen.

A constitutive representation for linear aging, environmental-dependent viscoelastic materials

Abstract: This study is concerned with developing a constitutive law for aging and environmental-dependent materials. The development rests on the assumption that the mechanical properties of the aforementioned materials can be mathematically represented by a functional of the strain and environmental histories, and this functional depends on the present time and the time the material is created. This constitutive assumption leads to two equivalent integral forms of the constitutive law after asserting that the functional is linear in the strain history. The first form of the integral law contains a material response functional analogous to the creep and relaxation functions of classical linear viscoelasticity. The second integral law has the same basic mathematical form but the physical interpretation of the mechanical response functional is different. It is demonstrated that both forms of the aging law reduce to the same non-aging law which is the usual starting for an analysis of only environmental-dependent materials. In the Appendix an operator algebra is presented for convenience in manipulating the integral laws for aging materials.

Stress concentration around a cylindrical cavity in a bone treated as a poroelastic body

Abstract: Two-phase poroelastic material is taken as a model of a living bone in the sense that the osseous tissue is treated as a linear isotropic perfectly elastic solid, and the fluid substances filling the pores as a Newtonian viscous fluid. Using Biot equations, derived in his consolidation theory, and assuming a plane state of strain governing equations involving fluid excess pressure and the stress function are derived. Laplace transform technique enables one to find explicit solutions for stresses. It is found that under a constant external load the bone element starts to creep, so that the viscoelastic properties of the adopted bone model seem to be in agreement with the experimental findings ofSedlin.

Dynamics of polar fluids

Abstract: The theory of polar fluids is discussed from a physical standpoint. Field equations and constitutive equations are established from a different point of view. Using the entropy inequality, restrictions on material coefficients are obtained. Various boundary conditions are discussed and a new boundary condition is suggested. The classical slow motion concept is extended to the case of polar fluid motion. The expression of the increased rate of dissipation in an incompressible suspension is given and the viscosity of a dilute suspension is calculated. A convenient expression for the force exerted by the fluid on a body is established and the expression of the force by fluid on a rigid sphere in uniform translation is obtained.

On Hill's stress rate in the continuum mechanics of polycrystals

Abstract: The lattice corotational stress rate, suggested byHill [1], is contrasted with the “material” co-rotational (Jaumann-Zaremba) stress rate and shown to be preferred in phenomenological continuum models of crystal behavior. A simple application to a crystal deforming via quasi-static single slip is included.

Nachweis der Eigenwerte 0 und −1 in den Spektren von Integraloperatoren der ebenen Elastizitätstheorie

Abstract: In den Spektren von Integraloperatoren der ebenen Elastizitatstheorie lassen sich fur spezielle Scheibenformen mit Hilfe gezielter Ansatze isolierte Eigenwerte 0 und −1 nachweisen. Die physikalische Bedeutung dieser Eigenwerte und der dazugehorigen Eigenfunktionen last vermuten, das die Eigenwerte auch in den Spektren der Operatoren fur beliebig geformte Scheiben auftreten. In dieser Arbeit wird bewiesen, das die Vermutung richtig ist.

Thermal stresses in a thin circular disc of orthotropic material due to an instantaneous point heat source

Abstract: In this paper, we have analysed the transient plane thermal stress problem of a circular disc of orthotropic material with instantaneous point heat source. The variation of σθθ with time along different radius vectors is exhibited graphically and compared with that of the isotropic case.

On the identification problem for thermorheologically simple materials

Abstract: In this paper, an identification problem in the theory of thermorheologically simple materials is discussed. An optimization procedure for estimation of the shift function and characteristic master curve is developed using ideas of differential approximation and quasilinearization. The nature and number of experimental stress and strain history pairs used for identification purposes is not restricted, but the procedure is limited to isothermal tests, a condition generally met in practice. Two methods, depending on the nature of available data, applicability of the shift postulate, and other considerations are presented. In both instances the characteristic master curve is finally given as the solution of a suitable differential equation subject to initial conditions, a convenient functional parameterization if a digital computer is to be used in subsequent stress analysis. Finally, a numerical example is presented to illustrate the feasibility and accuracy of the method.

Coupled theory of thermoelastic plates

Abstract: In this paper the coupled thermoelastic theory of a thin plate is discussed. Using a perturbation method, the free vibration frequencies of plates under various boundary conditions are found. Furthermore, a solution is given for the problem of a generally loaded thermoelastic plate and thermoelastic dissipations are shown.

Nearly isochoric finite torsion of a compressible isotropic elastic circular cylinder

Abstract: Finite torsion of a circular bar of isotropic compressible hyperelastic material is considered. A procedure suggested by Truesdell is used to obtain solutions for a particular strain energy function although the method used is not restricted to this particular form. The procedure is applicable if the volume strain is small. Results for finite twist with the length prevented from changing and with the length allowed to change so that the resultant longitudinal force is zero are presented.

Einfluß einer kleinen Biegesteifigkeit auf die Querschwingungen einer eingespannten rechteckigen Membran

Abstract: Das Eigenwertproblem fur die Biegeschwingungen einer in ihrer Mittelebene gleichmasig gezogenen Rechteckplatte wird behandelt unter der Voraussetzung, das ein dimensionsloser Parameter e, der ein Mas fur die relative Bedeutung der Biegesteifigkeit im Vergleich zur Vorspannung ist, wesentlich kleiner als eins sei. Das singulare Storproblem wird mit der Methode der verknupften asymptotischen Entwicklungen gelost. In erster Ordnung ergeben sich erwartungsgemas die Eigenwerte (Eigenfrequenzen) einer idealen (vollkommen biegeschlaffen) Membran; die Losung zweiter Ordnung jedoch last den Einflus einer kleinen Biegesteifigkeit auf die Eigenfrequenzen der eingespannten Membran erkennen.

Stress functions in three-dimensional elastodynamics

Abstract: After a historical sketch concerning the introduction of potential functions in elastodynamics, and its basic equations, a representation in terms of stresses is given with the aid of six stress functions, which are components of a symmetric stress tensor, of the second order (Schaefer's tensor) and satisfy the equation of transverse wave propagation, and with the aid of another stress function. It is also shown that this representation is equivalent to the representations given byM. Iacovache [1] andE. Sternberg andR. A. Eubanks [2] and that it is complete (each state of displacement and stress corresponding to, the problem can be expressed in this form) for a simply connected region. The case of arbitrary body forces as well as the case of conservative body forces are also considered. The corresponding particular integrals are obtained in the form of distributions with the aid of integral transforms of the Laplace and Fourier type. The results can therefore be applied to the case of concentrated loads. In particular, the case of a concentrated force as well as the case of a concentrated nucleus of space dilatation are treated. The latter case can be expressed in terms of conservative body forces.

Stability in a linear theory of elastic rods

Abstract: A linear dynamical theory of elastic rods is formulated from a theory of one-dimensional oriented continua. Sufficient conditions for stability, uniqueness and instability of a rod under applied dissipative forces and couples are given.

Zur Lösung nichtlinearer stochastischer Probleme der Mechanik

Abstract: Diese Arbeit beschaftigt sich mit dem Problem der Ermittlung der Momente des Zustandsvektors nichtlinearer, stochastisch erregter, mechanischer Systeme. Ausgangspunkt der vorgeschlagenen Methode ist die Fokker-Planck-Gleichung, die Differentialgleichung der Ubergangswahrscheinlichkeitsdichte. Bedingungen fur die Abgeschlossenheit im Endlichen der von dieser Differentialgleichung abgeleiteten Systeme von Differentialgleichungen fur die Momente des Zustandsvektors, eine Methode zum naherungsweisen Abschlus im Endlichen und zwei einfache Beispiele werden angegeben.

On the mechanics of materials with microstructure

Abstract: Considered is a material body composed of microelements or particles. The particles are represented by a point and by a number of directors. Under suitable assumptions such a discrete medium is approximated by a generalized Cosserat continuum with deformable directors. Differential equations of motion are derived from the first law of thermodynamics using an extension of the invariance principle proposed byGreen andRivlin. The number of the equations of motion corresponds to the number of directors and the system of the obtained equations may be regarded as a complete system. As an application the constitutive equations for elastic materials with microstructure are derived.

Rolling Contact of a Rigid Cylinder Over a Smooth Elastic or Viscoelastic Layer

Abstract: The rolling contact of a rigid cylinder over a smooth elastic or viscoelastic layer is investigated. Steady rolling is assumed and inertia terms are neglected so slow rolling in some sense is implied. The boundary-value problem is formulated for a finite spectrum viscoelastic model and illustrated by a standard linear solid. A singular integral equation is derived for the contact pressure, whose kernel is approximated to high accuracy by a logarithm multiplied by a polynomial plus a second polynomial. The latter integral equation is solved approximately and accuracy of solution verified. A geometrical parameter β defining the ratio of layer thickness to cylinder arises, and valid solutions within the framework of infinitesimal deformation gradient are obtained down to β=10−3, classified as a thin layer.

Acceleration waves in anisotropic thermoelastic materials with internal state variables

Abstract: This work is concerned with the effect of internal state variables on the growth and decay properties of acceleration waves in anisotropic nonlinear thermoelastic materials. Sufficient conditions are given which insure the existence of a plane longitudinal acceleration wave and two transverse acceleration waves with orthogonal amplitudes.

Source in a rotating stratified fluid

Abstract: In this paper solutions are presented in closed form for the flow due to a point source of oscillatory strength in an inviscid, rotating, stratified liquid. The solutions do not exhibit any spherical symmetry. The flow due to a steady source is deduced as a limit.

On the motion of a curved and twisted rod

Abstract: In this paper, the equations of motion of a curved and twisted rod are derived from the basic principles of dynamics. The set of equations gives the extensional, flexural and torsional motions of the rod. The coupling among these types of motion due to the curvature and tortuosity of the rod is shown explicitly in the case of a helical spring. By successive simplification of the equations, the equations of motion of a circular ring and those of a straight rod are obtained. In this respect, the derived equations can be considered as a generalization of the elementary theories of rod in extensional, torsional and flexural motion.