Academic literature on the topic 'Prandtl-Reuss plasticity'

We study the n-dimensional wave equation with an elastoplastic nonlinear stress–strain relation. We investigate the case of heterogeneous materials, i.e., x-dependent parameters that are periodic at the scale η>0. We study the limit η→0 and derive the plasticity equations for the homogenized material. We prove the well-posedness for the original and the effective system with a finite-element approximation. The approximate solutions are also used in the homogenization proof which is based on oscillating test functions and an adapted version of the div-curl Lemma.

The occurrence of small scale plasticity can be modeled physically by force doublets embedded in an elastic medium and therefore the plasticity problem can be treated by the superposition of elastic solutions. This idea for the treatment of an inelastic strain is reviewed and generalized to develop a versatile program for two-dimensional elastic-plastic problems based on Body Force Method. In the present study, a treatment of an elastic-perfect plastic body is discussed in detail. The increment of the density of force doublets, which has one to one correspondence to the increment of plastic strain, can be determined from Prandtl-Reuss equation. It was also found the Delaunay triangulation is useful and convenient for the automated elastic-plastic analysis.

We consider the quasi-static evolution of the thermo-plasticity model in which the evolution equation law for the inelastic strain is given by the Prandtl–Reuss flow rule. The thermal part of the Cauchy stress tensor is not linearized in the neighbourhood of a reference temperature. This nonlinear thermal part is imposed to add a damping term to the balance of the momentum, which can be interpreted as external forces acting on the material. In general, the dissipation term occurring in the heat equation is an integrable function only and the standard methods can not be applied. Combining truncation techniques and Boccardo-Gallouët approach with monotone methods, we prove an existence of renormalized solutions.

We present a finite element implementation of a Cosserat elasto-plastic model and provide a rigorous numerical analysis of the introduced time-incremental algorithm. The model allows the use of standard tools from convex analysis as known from classical Prandtl–Reuss plasticity. We derive the dual stress formulation and prove that for vanishing Cosserat couple modulus μc → 0 the classical problem is approximated. Our numerical results show the robustness of the approximation. Notably, for positive couple modulus μc > 0 there is no need for a safe-load assumption. For small μc the response is numerically indistinguishable from the classical response.

The so-called “plastic buckling paradox” originates from the fact that the Prandtl–Reuss’ flow theory of plasticity overestimates the plastic buckling load of plates and shells, whereas Hencky’s deformation theory of plasticity provides results that are more accurate. However, it has been shown that this problem can be overcome by introducing certain initial imperfection in accurate finite element (FE) simulations based on the flow theory of plasticity. 1 – 4 The present study goes deeper into the problem and reveals that in the case of short cylinders under combined loading, which have long been the object of extensive research in the elastic range, 5 a different modeling of the material behavior can also trigger a mode jumping from the initial imperfection, which may even reverse the reported predictions by the flow and deformation theories of plasticity. This fact must be taken in maximum consideration when performing nonlinear FE analyses for estimating the plastic buckling of thin and moderately thin short cylindrical shells.

In this work we develop a rigorous mathematical analysis of variational problems describing the quasistatic evolutionary problems in plasticity. The common feature of the problems under consideration is the variational energy formulation, where the mathematical difficulties arise due to the presence of a term with a linear growth in the symmetric part of the gradient of the unknown vector-valued functions in a volumetric case, or on the Hessian of the unknown scalar function in two-dimensional problems for plates. In the applications these functions represent displacement fields of a body...

The ability to predict fatigue crack growth rates is essential in safety critical systems. The discovery of fatigue crack closure in 1970 caused a flourish of research in attempts to simulate this behaviour, which crucially affects crack growth rates. Historically, crack tip plasticity models have been based on one-dimensional rays of plasticity emanating from the crack tip, either co-linear with the crack (for the case of plane stress), or at a chosen angle in the plane of analysis (for plane strain). In this thesis, one such model for plane stress, developed to predict fatigue crack closure, has been refined. It is applied to a study of the relationship between the apparent stress intensity range (easily calculated using linear elastic fracture mechanics), and the true stress intensity range, which includes the effects of plasticity induced fatigue crack closure. Results are presented for all load cases for a finite crack in an infinite plane, and a method is demonstrated which allows the calculation of the true stress intensity range for a growing crack, based only on the apparent stress intensity range for a static crack. Although the yield criterion is satisfied along the plastic ray, these one-dimensional plasticity models violate the yield criterion in the area immediately surrounding the plasticity ray. An area plasticity model is therefore required in order to model the plasticity more accurately. This thesis develops such a model by distributing dislocations over an area. Use of the model reveals that current methods for incremental plasticity algorithms using distributed dislocations produce an over-constrained system, due to misleading assumptions concerning the normality condition. A method is presented which allows the system an extra degree of freedom; this requires the introduction of a parameter, derived using the Prandtl-Reuss flow rule, which relates the magnitude of slip on complementary shear planes. The method is applied to two problems, confirming its validity.

L’étude du comportement de B. Linens en fermenteur est entreprise sur un milieu à base de L-lactate et d’une peptone caséique. La première partie présente la réalisation d’un plan d’expériences pour mettre en évidence l’effet de 3 facteurs, le pH (7,5-8,5), la température (20-30°C), et l’oxygène dissous (40-60% de la saturation), tous 3 régulés au cours de la culture, sur plusieurs paramètres de croissance et de métabolisme à des temps caractéristiques : temps de latence, temps de fin de phase exponentielle, temps de fin de phase de ralentissement et temps final. La valeur de ces temps, la densité optique (D. O. ) de la culture, la quantité de lactate et d’acides aminés demeurant présents, ainsi que l’ammoniac produis à ces temps sont analysés. Par ailleurs, la valeur du temps de génération est déduite de représentations graphiques. Un modèle logistique de croissance a été également utilisé pour une meilleure précision, et les paramètres suivants ont été déduits : temps pour attendre 2. 5, 35, et 80% de la D. O. Maximale, valeur ajustée de ce maxima, vitesse maximale et moyenne de croissance. Les données fournies par les régulations de pH et d’oxygène dissous ont été également analysées aux mêmes temps caractéristiques. Les résultats les plus satisfaisants sont obtenus par l’analyse de variance sur l’ensemble des conditions expérimentales et le paramètre temps de génération. Ils révèlent un important effet quadratique de l’oxygène, une liaison très importante entre les 3 facteurs, et des termes tous positifs. Le temps de génération diminue entre 40 et 50% d’oxygène et augmente au-delà. Les variations des autres paramètres sont moins bien expliquées, mais il ressort que la valeur maximale de la D. O. Est indépendante des niveaux des facteurs. La disparition des substrats n’indique pas d’effet de facteurs compte-tenu du manque de précision des mesures, sauf un effet linéaire, quadratique, et cubique de l’oxygène : ses niveaux 40 et 60% entraînent une consommation incomplète des substrats par comparaison au niveau 45-55% mais la croissance est-elle achevée dans ces cultures ? B. Linens produit de l’ammoniac, qu’il reconsomme, apparemment lorsque la concentration d’acides aminés présents diminue, ceci étant révélé par des régressions statistiques significatives. Il consomme préférentiellement certains acides aminés (arginine, tyrosine, phénylalaline) et produit dans certains cas de l’ornithine. Ses activités de production d’arômes sont mises en évidence qualitativement par des reniflages. L’utilisation d’un inoculum de cellules lyophilisées a entraîné l’allongement des temps de latence, et une consommation accrue en oxygène. Les quantités de neutralisant semblent dépendre plus des taux de croissance que des conditions de culture. Quelques expérimentations supplémentaires réalisées avec des milieux de concentrations différentes, et présentées en partie 2 ont indiqué que : - le lactate n’est pas limitant – certains acides aminés préférentiellement consommés doivent être limitants, et leur disparition entraîne la chute rapide des taux de croissance – l’ammoniac est produit lorsque la source azotée est présente en quantité supérieure à la source carbonée – les rendements faibles en biomasse dans les cas de milieux à forte concentration en lactate peuvent s’expliquer par un terme de maintenance. Quelques essais de modélisation de la production de biomasse et de disparition des substrats sont réalisés, mais la complexité des milieux employés empêche de conclure
This study presents the behaviour of Brevibacterium linens in a fermentor on a L-lactate and casamino acids medium. The first part consists of an experimental design used to point out the effect of 3 factors: the pH (7,5-8,5), the temperature (20-30°C) and the dissolved oxygen concentration (40-60% of the saturation), both of them controlled during the culture, on growth and metabolism parameters at different times: lag time, end of exponential growth, end of the falling back of growth and final time. The values of these times, the absorbance (D. O. ) of the culture, the concentration of lactate and amino acids, and produced ammonia at these times were analysed. The value of generation time was deduced from graphical response. The logistic law was used to improve growth data precision, and the following parameters were deduced: time to reach 2. 5, 35, and 80% of the maximal O. D. , value of this maximum, maximal and mean growth rate. Data obtained from the pH and dissolved oxygen controls at the same typical times were also analysed. The most important results were obtained from the variance analysis on the overall experimental conditions and on the generation time parameter: they showed an important quadratic effect of oxygen, an important interaction between the 3 factors, and positive effects. The generation time decreased between 40% and 50% of dissolved oxygen and increased beyond this level. The other parameters were less explained, but it appeared that the value of maximal growth is factors independent. The substrate consumption pointed out no factor effect, with the obtained precision, except a linear, quadratic and cubic effect of dissolved oxygen: tested at the levels 40 and 60%, this factor leads to incomplete substrate consumption, compared with the level 45-55%. But, is the maximal growth achieved in the case of these cultures? B. Linens produced ammonia which is absorbed with the amino acids, this fact pointed out by linear regressions statistically significant. A preferential consumption of a few amino acids (arginine, tyrosine and phenylalanine) has been detected, as ornithine production in some circumstances. His aroma production activities were pointed out qualitatively by sniffing. The freeze dryed suspension showed longer lag periods, and a greater oxygen consumption. The volumes of acid or alkali used to maintain the pH seemed to be more dependent on growth than on culture conditions. Some additional experiments presented in part 2, with mediums of different substrate concentrations showed that: - lactate is not limitant – some amino acids absorbed may be limitant, and decrease in growth rate may be explained by their removing – the low growth yields found in case of high lactate concentration medium may be explained by maintenances effects. Modelisation of biomass production and substrate consumption was tested, but the medium complexity prevented to do it

Abstract The small-deformation theory of elastic-plastic response, due to Prandtl and Reuss, is the basis of most analytical treatments of elastic-plastic behavior. This model is examined from a modern perspective. The shifter concept is combined with the convected coordinate formalism to cast the theory in a manner that ensures its invariance under a change of frame, in contrast to conventional expositions of the small-deformation theories of elasticity and elasto-plastic response

This chapter provides an introduction to isotropic hardening plasticity models in the three-dimensional small strain setting. The additive decomposition of the strain is introduced along with the concepts of plastic strain and equivalent tensile plastic strain for three-dimensional problems. Plastic flow is discussed and defined, and a complete model of plasticity is formulated with Kuhn-Tucker loading/unloading conditions. Both rate independent and rate dependent (viscoplastic) models are discussed with an emphasis on Mises-Hill type theories which utilizes a Prandtl-Reuss flow rule. A variety of important flow models are presented.

The values of residual stresses resulting from a heavily stressed contact are numerically evaluated by employing a three-dimensional strain deformation analysis model. The model is developed in the frame of the incremental theory of plasticity including the von Mises yield criterion, Prandtl-Reuss equations, and Ramberg-Osgood stress-strain equation. Two groups of cylindrical specimens were subjected to rolling contact fatigue, one as the reference group and the other with an induced residual stresses state. To obtain theoretical lives of the tested groups the von Mises equivalent stress is used in Ioannides-Harris rolling contact fatigue model. Both, the experimental data and theoretical analysis reveal more than two times greater fatigue life for the group with induced residual stresses versus the life of the reference group.

An analysis model has been developed to model the nonlinear strain rate dependent deformation of rolling bearing steel stressed in the elastic-plastic domain. The model is developed in the frame of the incremental theory of plasticity by using the von Mises yield criterion and Prandtl-Reuss equations. By considering the isotropic and non-linear kinematic hardening laws of Lemaitre-Caboche, the model accounts for the cyclic hardening phenomena. To attain the final load of each loading cycle, the two bodies are brought into contact incrementally. For each new load increment new increments for the components of stress and strain tensors, but also increments of residual stresses, are computed for each point of the 3D mesh. Both, the new contact geometry and residual stresses distributions, are further considered as initial values for the next loading cycle, the incremental technique being reiterated. The cyclic evaluation process of both, plastic strains and residual stresses is performed until the material shakedowns. The experimental part of the paper regards to the rolling contact fatigue tests carried out on two groups of line contact test specimens and on two groups of deep groove ball bearings. In both cases, the experimental data reveal more than two times greater fatigue life for the group with induced residual stresses versus the life of the reference group. The von Mises equivalent stress is considered in Ioannides-Harris rolling contact fatigue model to obtain theoretical lives. The theoretical analysis revealed greater fatigue lives for the test specimens and for the ball bearings groups with induced residual stresses than the fatigue lives of the corresponding reference groups.

The fatigue life tests carried out on two groups of ball bearings confirm the positive influence of the compressive residual stresses induced by a previous loading in the elastic-plastic domain. The values of residual stresses are numerically evaluated by employing a three-dimensional strain deformation analysis model. The model is developed in the frame of the incremental theory of plasticity by using the von Mises yield criterion and Prandtl-Reuss equations. To consider the material behaviour the Ramberg-Osgood stress-strain equation is involved and a nonlinear equation is considered to model the influence of the retained austenite. To attain the final load of each loading cycle the two bodies are brought into contact incrementally, so that for each new load increment the new pressure distribution is obtained as the solution of a constrained system of equation. Conjugate gradients method in conjunction with discrete convolution fast Fourier transform is used to solve the huge system of equations. Both the new contact geometry and residual stresses distributions, are further considered as initial values for the next loading cycle, the incremental technique being reiterated. The cyclic evaluation process of both plastic strains and residual stresses is performed until the material shakedowns. Comparisons of the computed residual stresses and deformed profiles with corresponding measured values reveal a good agreement and validate the analysis model. The von Mises equivalent stress, able to include both elastic and residual stresses, is considered in Ioannides-Harris rolling contact fatigue model to obtain theoretical lives of the ball bearings groups. The theoretical analysis reveals also greater fatigue lives for the ball bearings groups with induced residual stresses than the fatigue lives of the group without induced residual stresses.

The roller profile appears to be the key element to attain a longer rating life for both cylindrical and tapered roller bearings. A genuine elastic analysis is able to optimize the roller profile to obtain a stress distribution in the contact zones that provides enhanced operational reliability and greater insensitivity to misalignment. For traditional cylindrical-crowned roller profile design class I discontinuities exist at the intersection points of roller profile with the crowning radius as well as at the end chamfer. In an elastic analysis these discontinuities generate very sharp increases in pressure distribution diminishing the rating life of the bearing. In fact, these local increases in pressure distribution are able to overcome, locally, the yield limit and to induce both plastic deformations and residual stresses. After a certain number of cycles the material will shakedown elastically to a slightly modified roller profile and a stable state of compressive residual stresses. If were taken place, these changes have to be considered in the life evaluation. An analysis model has been developed to simulate the nonlinear strain rate dependent deformation of rolling bearing steel stressed in the elastic-plastic domain. The model is developed in the frame of the incremental theory of plasticity by using the von Mises yield criterion and Prandtl-Reuss equations. By considering an isotropic and non-linear kinematic hardening laws the model accounts for the cyclic hardening phenomena. For each new load increment new increments for the components of stress and strain tensors, but also increments of residual stresses, are computed for each point of the 3D mesh. Both the new contact geometry and residual stresses distributions, are further considered as initial values for the next loading cycle, the incremental technique being reiterated. The cyclic evaluation process of both the plastic strains and residual stresses is performed until the material shakedowns. For the case of cylindrical roller bearings with cylindrical-crowned roller profile, the role played by the crowning geometry on pressure distribution is pointed out for both the elastic analysis and elastic-plastic analysis. Further, the modified rating lives are evaluated using the methodology given in ISO 16281-2008.