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Phenomenological Modelling of Viscoplasticity E Phenomenological modelling of viscoplasticity E. Krempl To cite this version: E. Krempl. Phenomenological modelling of viscoplasticity. Revue de Physique Appliquée, Société française de physique / EDP, 1988, 23 (4), pp.331-338. 10.1051/rphysap:01988002304033100. jpa- 00245778 HAL Id: jpa-00245778 https://hal.archives-ouvertes.fr/jpa-00245778 Submitted on 1 Jan 1988 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Revue Phys. Appl. 23 (1988) 331-338 AVRIL 1988, 331 Classification Physics Abstracts 62.20 Phenomenological modelling of viscoplasticity E. Krempl Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A. (Reçu le 26 mai 1987, révisé le 31 août 1987, accepté le 15 septembre 1987) Résumé.- Les bases de la modélisation phénoménologique du comportement des métaux soumis à de petites dé- formations sont introduites avec les expériences effectuées à l’aide de machines d’essai servocontrollées et des mesures de déformation fournies par les jauges sur les éprouvettes. L’expérimentation systématique conduit à la théorie de la viscoplasticité basée sur l’excès de contrainte, et ses propriétés sont déter- minées. Avec le chargement à vitesse de déformation constante, la théorie admet les solutions asymptoti- ques qui sont relatives à l’écoulement plastique pur dans l’expérimentation. Une nouvelle mesure de la sen- sitivité de vitesse ne dépendant pas du niveau de contrainte est proposée. Les essais de relaxation, qui étaient interprétés du point de vue de la science des matériaux, sont réanalysés du point de vue phénomé- nologique avec des conclusions différentes. Abstract. - The essentials of phenomenological modeling of metal deformation behavior at small strain are introduced together with companion experiments which are performed with servocontrolled testing machines and strain measurement on the specimen gage length. Systematic experimentation leads to the viscoplastic- ity theory based on overstress and its properties are delineated. The theory admits asymptotic solutions under constant strain rate loading and they are related to fully established plastic flow in the experiment. A new measure of rate sensitivity is proposed which does not depend on the stress level. Relaxation exper- iments which were interpreted from a materials science viewpoint are re-analyzed from a phenomenological point of view with different conclusions. INTRODUCTION are force-displacement pairs. One of the quanti- ties (force or displacement) is usually prescribed In this paper a continuum mechanics, phenomenolog- as a function of time, and it is called the input; ical viewpoint is adopted. The postulates of a the other (displacement or force) represents the representative volume element and of homogeneous output, the answer of the material to the input. states of stress (strain) are fundamental. It is The study of input-output pairs gives information recognized that matter consists of atoms, molecules, on the material and how it changes with deformation. dislocations and other discrete particles. The aim It is the premise of a phenomenological approach is not to describe their motions; rather the aim is that a suitable combination of input-output pairs to capture the macroscopic deformation behavior of is sufficient to obtain the essential features of metallic materials in a mathematical model, the con- the material deformation behavior in a domain of stitutive equation. In this case, a "smeared-out" interest. Continuum mechanics provides tne basic description is adopted. conservation laws of nature and the methods of To accomplish this goal the existence of a volume reducing forces and displacements to proper stress element is postulated which contains a sufficient and strain measures, respectively. It further number of the microstructural, discrete elements so generalizes the behavior and provides for predic- that its response is representative of the material tive capabilities. under investigation. This means that the size of In materials sciences, macroscopic experiments the volume element has to contain several grains if similar to the ones used by the phenomenologists a polycrystalline material is considered. Within are used to delineate appropriate microstructural this volume the state of stress (strain) is con- mechanisms which are then augmented by other tests, sidered to be homogeneous or uniform. A further such as electron microscopy and x-ray analysis to assumption is that the material itself has homo- name just a few (Kocks, Argon and Ashby [1]). geneity so that the properties of the representa- Since the material scientists and the mechani- tive volume element do not depend on the location cians use tests on a macroscopic sample for their within a material. studies, such testing would constitute a natural In many cases the volume element is a test speci- starting point of much needed cooperation. The men, frequently a cylindrical or a tubular bar, and results could be interpreted by each discipline the basic information obtained from the experiments separately and then discussed jointly. Everyone Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304033100 332 could benefit from such a joint program. stress," and relaxation evolves when "constant strain" is imposed, the two phenomena are separated THE PHENOMENOLOGICAL APPROACH in a continuum approach. It is recognized that the evolution of the creep or the relaxation curve is The deformation behavior of engineering alloys used governed by the "creep properties" of the material in structural applications at homologous tempera- at a given temperature but they manifest themselves tures up to 0.6 and strain rates up to 10-2 1/s is differently under different boundary conditions. of The strains are small and are encoun- interest. Figure 2 gives an example of history dependence tered in monotonic and cyclic loadings possibly for the strongly hardening type 304 stainless steel involving periods of creep or relaxation. Tran- at room temperature. A virgin specimen and a spec- sients are frequent and the assumption of a steady- imen, which underwent completely reversed strain state behavior is rarely realistic. The task is to controlled loading to cyclic saturation followed by establish a three-dimensional constitutive equation unloading to zero stress and strain, are subjected which captures the essential features of the macro- to the same input, strain rate cycling with two scopic deformation behavior, including Krempl [2]: orders of magnitude difference. It is seen that the stress-strain curve of the specimen with prior 1) Rate dependence which encompasses stress and cyclic loading is considerably higher than the strain rate sensitivity, creep and virgin stress-strain curve, and that the stress relaxation. level differences between the curve at the differ- ent rates are approximately the same for both 2) History dependence in the sense of plasticity Prior has not - sometimes called (path dependent) hardening. specimens. cyclic hardening appreciably changed the rate sensitivity but has This term does not refer to the positive significantly elevated the stress the mani- slope of a stress-strain diagram but rather level, festation of history dependence in the sense of describes a unique property of crystalline plasticity or of (path dépendent) hardening. solids. The response to the same input can The results of Fig.2 suggest that the rate sensi- be substantially different depending on prior tivity is, as first unchanged history. It includes such phenomena as approximation, by cyclic hardening. The measure of rate hardening or softening. An example sensitivity cyclic used in materials science [1] is will be given below. 3) Recovery, the reversal of hardening by the action of diffusion. 4) Aging, the change of mechanical properties in the absence of mechanical deformation. Aging where ÿ and o are the inelastic strain rate and the can be caused by diffusion and chemical reac- current stress, respectively, and where the sub- tions within the material, or by exchange of script T indicates that this quantity is to be matter with the environment and subsequent determined at constant temperature. chemical reactions. If (1) is evaluated for the conditions of Fig.2, i.e. for the virgin stress-strain curve (sub- The phenomena delineated above (except 3) are script 1) and for the curve after prior cycling operationally defined by Krempl [2J and can be de- (subscript 2), with the assumption that the stress materials tests. tected in real through macroscopic level differences are the same for 1 and 2, then the will be on items 1) In this paper emphasis ml/m2= Ql/Q2 where a denotes the stress level. that is absent. and 2) and it will be assumed aging Since the stress levels in Fig.2 are roughly dif- is another form of It is realized that there aging, ferent by a factor of two, the m values differ by and strain aging which involves diffusion processes the same factor. This result is at variance with chemical reactions triggered
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