Some theoretical considerations and experimental results concerning elastic-plastic stress-strain relations

Summary Within the frame of the classical phenomenological theory of plasticity the yield condition and the inelastic stress-strain relations are very restrictively correlated by the so-called normality rule. For certain classes of problems this leads to an unsatisfactory agreement with experimental results. It can be shown how the classical theory can be generalized in a suitable manner without violating the fundamental thermo-dynamical laws in order to obtain a better agreement between theoretical and experimental results in problems with non-proportional loading history.

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Einige theoretische Betrachtungen und experimentelle Untersuchungen zum elasto-plastischen Formänderungsgesetz

Übersicht Im Rahmen der klassischen phänomenologisehen Plastizitätstheorie ist das Formänderungsgesetz, d. h. die Beziehungen zwischen den Spannungs- und den Verzerrungsinkrementen durch die sog. Normalenregel, sehr eng mit der Fließbedingung verknüpft. Bei gewissen Problemen führt dies zu einer unbefriedigenden Übereinstimmung mit experimentellen Befunden. Es wird gezeigt, wie die klassische Theorie in geeigneter Weise verallgemeinert werden kann, ohne zu Widersprüchen im Rahmen der Thermodynamik zu führen. Auf diesem Wege läßt sich eine bessere Übereinstimmung zwischen theoretischen und experimentellen Ergebnissen bei nicht-proportionalen Belastungsprozessen erzielen.

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This paper was presented at the Second Symposium on Inelastic Solids and Structures held in Bad Honnef in September 1981     

Sufficient uniqueness and stability conditions for elastic-plastic structures with associated flow laws

Summary The incremental problem for elastic-plastic structures with associated flow laws is stated in terms of an integral equation, whose characteristics and solution are discussed.It is then shown how a simpler problem whose condition of uniqueness of solution constitutes an easily applicable sufficient condition of uniqueness and stability of response for the incremental problem may be deduced.

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Sommario Si formula il problema incrementale per strutture elastoplastiche con leggi di scorrimento associate in termini d'una equazione integrale della quale si discutono le caratteristiche e il criterio di soluzione.Si mostra infine come dal problema incrementale possa essere dedotto un più semplice problema la cui condizione di unicità di soluzione costituisce per il problema incrementale una condizione sufficiente di unicità e stabilità della risposta, di semplice applicabilità.

     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.     

Techniques for measuring stress-strain relations at high strain rates

The qualitative dependence of the mechanical behavior of some materials on strain rate is now well known. But the quantitative relation between stress, strain and strain rate has been established for only a few materials and for only a limited range. This relation, the so-called constitutive equation, must be known before plasticity or plastic-wave-propagation theory can be used to predict the stress or strain distribution in parts subjected to impact stresses above the yield strength. In this paper, a brief review of some of the experimental techniques for measuring the stress, strain, strain-rate relationship is given, and some of the difficulties and shortcomings pointed out. Ordinary creep or tensile tests can be used at plastic-strain rates from 10^?8 to about 10^?1/sec. Special quasi-static tests, in which the stress- and strain-measuring devices as well as the specimen geometry and support have been optimized, are capable of giving accurate results to strain rates of about 10²/sec. At higher strain rates, it is shown that wave-propagation effects must be included in the design and analysis of the experiments. Special testing machines for measuring stress, strain and strain-rate relationships in compression, tension and shear at strain rates up to 10⁵/sec are described, and some of the results presented. With this type of testing machine, the analysis of the data requires certain assumptions whose validity depends upon proper design of the equipment. A critical evaluation of the accuracy of these types of tests is presented.     

Experimental dynamic analysis of elastic-plastic shear frames with secondary structures

Various experimental models are developed to study the influence of lightweight secondary structures on the dynamic response of elastic and elastic-plastic shear frames. Small-scale two-story model frames, with an elastic single-degree-of-freedom secondary structure attached, are considered for sinusoidal and random in-plane support excitation. Both elastic and elastic-plastic responses are recorded by varying the material properties of the columns of a distinguished floor. Parametric studies are performed by varying the secondary structure's fundamental frequency and damping. Experimental results are compared with those obtained by computational simulations. Experimental and numerical results are in excellent agreement, however they show that the material properties have to be determined very carefully. The statistic response of randomly excited elastic-plastic structures is not much affected by the motion of tuned secondary structures. However, this dynamic behavior is not true for elastic main structures. In this case, an optimally tuned secondary structure decreases the structural response up to 25%.     

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.     

On shakedown theory for elastic-plastic materials and extensions

The idea that an elastic-plastic structure under given loading history may shake down to some purely elastic state (and hence to a safe state) after a finite amount of initial plastic deformation, can apply to many sophisticated material models with possible allowable changes of additional material characteristics, as has been done in the literature. Despite some claims to the contrary, it is shown; however, that the shakedown theorems in a Melan-Koiter path-independent sense have been extended successfully only for certain elastic-plastic hardening materials of practical significance. Shakedown of kinematic hardening material is determined by the ultimate and initial yield stresses, not the generally plastic deformation history-dependent hardening curve between. The initial yield stress is no longer the convenient one (corresponding to the plastic deformation at the level of 0.2%) as in usual elastic-plastic analysis but to be related to the shakedown safety requirement of the structure and should be as small as the fatigue limit for arbitrary high-cycle loading. Though the ultimate yield strength is well defined in the standard monotonic loading experiment, it also should be reduced to the so-called “high-cycle ratchetting” stress for the path-independent shakedown analysis. A reduced simple form of the shakedown kinematic theorem without time integrals is conjectured for general practical uses. Application of the theorem is illustrated by examples for a hollow cylinder, sphere, and a clamped disk, under variable (including quasiperiodic dynamic) pressure.     

Method of substitutive plastic rotation applied to elastic-plastic structures with axial forces

Summary The paper extends the notion of substitutive plastic rotation which simplifies the determination of the elastic-plastic deflections of beams and frames to structures with non-negligible axial forces. The formulae derived have been illustrated with a simple example of a cantilever beam and then applied to the geometrically nonlinear elastic-plastic problem of an imperfection in a pipeline shell.

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Die Methode der Ersatzrotationen für elastisch-plastische Strukturen mit Axialkräften

Übersicht Die Arbeit erweitert die Methode der fiktiven Ersatzrotationen, die die Bestimmung der elastisch-plastischen Verschiebungen in Balken- und Rahmenstrukturen vereinfacht, auf den Fall mit größeren axialen Kräften. Die ausgeführten Formeln werden am einfachen Beispiel des Kragbalkens illustriert und dann für die geometrisch nichtlineare Analyse einer Beule in einer dünnwandigen Rohrleitung verwendet.

     

A microbeam bending method for studying stress-strain relations for metal thin films on silicon substrates

We have developed a microbeam bending technique for determining elastic-plastic, stress-strain relations for thin metal films on silicon substrates. The method is similar to previous microbeam bending techniques, except that triangular silicon microbeams are used in place of rectangular beams. The triangular beam has the advantage that the entire film on the top surface of the beam is subjected to a uniform state of plane strain as the beam is deflected, unlike the standard rectangular geometry where the bending is concentrated at the support. To extract the average stress-strain relations for the film, we present a method of analysis that requires computation of the neutral plane for bending, which changes as the film deforms plastically. This method can be used to determine the elastic-plastic properties of thin metal films on silicon substrates up to strains of about 1%.Utilizing this technique, both yielding and strain hardening of Cu thin films on silicon substrates have been investigated. Copper films with dual crystallographic textures and different grain sizes, as well as others with strong 〈1 1 1〉 textures have been studied. Three strongly textured 〈1 1 1〉 films were studied to examine the effect of film thickness on the deformation properties of the film. These films show very high rates of work hardening, and an increase in the yield stress and work hardening rate with decreasing film thickness, consistent with current dislocation models.     

Dynamic shakedown theory of elastoplastic work-hardening structures allowing for second-order geometric effects

This paper is based on piecewise linear yield surface and discretization of structure. By allowing for inertial force, damping force and second-order geometric effects, the two generalized dynamic shakedown theorems are given for shakedown analysis of structure.The project is supported by National Natural Science Foundation of China and Jilin Provincial Applying-Basic Research Projects.     

Procedures for construction of anisotropic elastic-plastic property closures for face-centered cubic polycrystals using first-order bounding relations

Microstructure-sensitive design (MSD) is a novel mathematical framework that facilitates a rigorous consideration of the material microstructure as a continuous design variable in the engineering design enterprise [Adams, B.L., Henrie, A., Henrie, B., Lyon, M., Kalidindi, S.R., Garmestani, H., 2001. Microstructure-sensitive design of a compliant beam. J. Mech. Phys. Solids 49(8), 1639-1663; Adams, B.L., Lyon, M., Henrie, B., 2004. Microstructures by design: linear problems in elastic-plastic design. Int. J. Plasticity 20(8-9), 1577-1602; Kalidindi, S.R., Houskamp, J.R., Lyons, M., Adams, B.L., 2004. Microstructure sensitive design of an orthotropic plate subjected to tensile load. Int. J. Plasticity 20(8-9), 1561-1575]. MSD employs spectral representations of the local state distribution functions in describing the microstructure quantitatively, and these in turn enable development of invertible linkages between microstructure and effective properties using established homogenization (composite) theories. As a natural extension of the recent publications in MSD, we provide in this paper a detailed account of the methods that can be readily used by mechanical designers to construct first-order elastic-plastic property closures. The main focus in this paper is on the crystallographic texture (also called Orientation Distribution Function or ODF) as the main microstructural parameter controlling the elastic and yield properties of cubic (fcc and bcc) polycrystalline metals. The following specific advances are described in this paper: (i) derivation of rigorous first-order bounds for the off-diagonal terms of the effective elastic stiffness tensor and their incorporation in the MSD framework, (ii) delineation of the union of the property closures corresponding to both the upper and lower bound theories resulting in comprehensive first-order closures, (iii) development of generalized and readily usable expressions for effective anisotropic elastic-plastic properties that could be applied to all cubic polycrystals, and (iv) identification of the locations of readily available or easily processable ODFs (e.g. textures that are produced by rolling, drawing, etc.) on the property closures. It is anticipated that the advances communicated in this paper will make the mathematical framework of MSD highly accessible to the mechanical designers.