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EN
The paper presents the analysis of electromechanical characteristics of piezoelectric converters subjected to an electric field and mechanical load. The analyses were performed based on a method consisting implementation of special segments responsible for electrical boundary conditions to a homogeneous beam. Constitutive equations were developed, allowing one to obtain static electromechanical characteristics for piezoelectric actuators with freely defined boundary conditions and geometry. Moreover, based on constitutive equations obtained, a particular solution for cantilever transducer subjected to concentrated force has been developed. The resulting analytical solution was compared with the data available in the literature, and the developed FEM solution. Furthermore, the influence of factors such as relative length, thickness and location of particular piezoelectric layers on electromechanical characteristics of the transducer was defined.
EN
The capabilities of different constitutive equations of approximating the tensile flow curves and correlating plastic behavior with the microstructure were investigated in austempered ductile iron ADI 1050. In a previous paper, the microstructure evolution of ADI 1050 during austempering was investigated through quenching the ADI 1050 after 14 increasing austempering times to room temperature. The 14 samples were tensile tested and two classes of constitutive equations were examined in the present paper. The Hollomon-type constitutive equations approximated all of the tensile flow curves of ADI 1050 very well but failed in correlating the plastic behavior with microstructure evolution. Voce-type constitutive equations approximated the tensile flow curves only at high stresses very well but could correlate the plastic behavior with the microstructure evolution of ADI 1050 during austempering excellently. The reason of this success was rationalized in terms of the physical basis of Voce-type equations, while Hollomon-type equations are empirical.
EN
The behavior of interfaces between granular soils and structural construction materials has an important impact on the monotonic and cyclic response of many soil–structure interaction (SSI) systems. Understanding the mechanics and modeling of these interfaces is an important step towards a safe and effective design and analysis of SSI problems. An extensive literature search for a ‘one-stop shop’ providing a comprehensive exposition on the mechanical characteristics and numerical modeling approaches of granular soil–structure interfaces yielded less than encouraging results. This paper seeks to bridge this knowledge gap in as concise manner as possible. To this end, the effects of the essential elements of the SSI problem, including soil and structural features as well as loading conditions, on the deformation and stress transformation mechanism of the interface are critically examined. Simple and advanced interface constitutive modeling methods are discussed, and implementation techniques of interface elements in finite element codes are explained. Additionally, a method to incorporate the effect of particle breakage to improve the capabilities of an elasto-plastic constitutive simulation of the cyclic accumulative contraction in granular interface modeling is introduced.
EN
In the case of a two-phase medium – such as the soil, which consists of an elastic skeleton and is filled with pore fluids – stress and strain within the medium are dependent on both phases. Similarly, in the case of heat transfer, heat is conducted through the two phases at different rates, with an additional heat transfer between the phases. In the classical approach to modelling a porous medium, it is assumed that the fluid filling the pore space is water, which is incompressible. In the case of gas, the volume of which is strongly dependent on temperature and pressure, one should take this behavior into account in the constitutive relations for the medium. This work defines the physical relations of a two-phase medium and provides heat transfer equations, constructed for a porous, elastic skeleton with fluid-filled pores, which may be: liquid, gas, or mixture of liquid and a gas in non-isothermal conditions. The paper will present constitutive relations derived from the laws of irreversible thermodynamics, assuming that pores are filled with either a liquid or a gas. These relations, in the opinion of the authors, may be used as the basis for the construction of a model of the medium filled partly with a liquid and partly with a gas. It includes the possibility of independent heat transfer through any given two-phase medium phase, with the transfer of heat between the phases.
PL
Konstytutywne równania dyfuzji wieloskładnikowej wyprowadzone w części pierwszej opracowania [1] w oparciu o mechaniczną teorię dyfuzji mogą być stosowane do dowolnych mieszanin izotropowych, takich jak gazy pod niskim i wysokim ciśnieniem, ciecze, roztwory elektrolitów i polimerów oraz plazma. Dlatego w tej części (drugiej) opracowania przedstawiono przykłady zastosowań zależności teoretycznych wyprowadzonych w części pierwszej do izotropowych układów płynnych wymienionych powyżej.
EN
The constitutive equations of multicomponent diffusion derived in the first part of the study [1] basing of the mechanical theory of diffusion are applicable to any isotropic fluid mixture like gases under low and high pressure, liquids, electrolyte and polymeric solutions as well as plasma. Therefore in this (second) part of the study examples of application of the theoretical expressions derived in the first part are presented with respect to the fluid systems mentioned above.
EN
This paper presents test results for deformation conditions of three-layer, piezoelectric cantilever converters subjected to various electrical and mechanical boundary conditions. A general solution has been developed based on implementation of piezoelectric triple segments (PTS) to the beam. A working mechanism and conditions for strain of the PTS segment have been determined. Basing on the general solution, for the cantilever actuator subjected to an external force (of single and dual PTS segments) and a uniform load (of single PTS segment), particular solutions have also been developed. Moreover, dimensionless frequencies of the oscillating motion for the analyzed converters have been determined. In the next step, the influence of such factors as length, quantity and position of PTS segments, their relative stiffness and unit weight on values of the obtained frequencies of vibration have been defined. The resulting analytical solutions have been compared with the developed FEM solution.
7
Content available remote Endochronic model of plasticity generalizing Sanders's theory
EN
The Sanders’s theory of plasticity and quasi-statistical variant of incremental plastic theory with isotropic and kinematic hardening in Novozhilov’s version are generalized in the frameworks of the endochronic approach. The constitutive equations of endochronic theory of inelasticity including the ideas of Sanders, Novozhilov and Valanis are formulated. The relations for the calculation of stresses and strains in uniaxial active and reversible material loadings are proposed. The formulas are obtained by using the elementary average principle of local values and by the simplest set of material constants and functions. Two types of initial conditions are considered in the calculations. The results of numerical modeling of inelastic material behavior under uniaxial active and cyclic loadings are presented. The results are compared with each other and with original Sanders’s theory. The similarity and differences between the generalized endochronic theory and the Sanders’s version are demonstrated. Several unusual manifestations of inelastic material behavior that require further theoretical analysis, calculations on the complex loading paths and the experimental verification are noted.
PL
W pracy w ramach podejścia endochronicznego dokonano uogólnienia teorii plastyczności Sandersa i quasi-statystycznego wariantu przyrostowej teorii plastyczności dla izotropowego kinematycznego umocnienia Nowo żyłowa. Sformułowano równania konstytutywne endochronicznej teorii zachowania niesprężystego, oparte na koncepcjach Sandersa, Nowo żyłowa i Valanisa. Zaproponowano zależności obliczeniowe dla naprężeń i odkształceń w przypadku aktywnego jednoosiowego obciążenia odwracalnego. Zależności te otrzymano przez zastosowanie zasady uśredniania wartości lokalnych dla podstawowego zestawu parametrów materiałowych. W obliczeniach rozważano dwa typy warunków początkowych. Zaprezentowano wyniki obliczeń numerycznych dla materiału niesprężystego poddanego aktywnemu obciążeniu jednoosiowemu oraz obciążeniu okresowemu. Dokonano analizy porównawczej wyników, w tym z wykorzystaniem teorii Sandersa. Przedstawiono różnice i podobieństwa pomiędzy uogólnioną teorią endochroniczną a teorią Sandersa. Wyszczególniono wiele niestandardowych efektów niesprężystego zachowania materiału, wymagających dalszych badań teoretycznych.
EN
The form of incremental constitutive equations for granular soils is discussed for the triaxial configuration. The classical elasto-plastic approach and the semi-empirical model are discussed on the basis of constitutive relations determined directly from experimental data. First, the general structure of elasto-plastic constitutive equations is presented. Then, the structure of semiempirical constitutive equations is described, and a method of calibrating the model is presented. This calibration method is based on a single experiment, performed in the triaxial apparatus, which also involves a partial verification of the model, on an atypical stress path. The model is shown to give reasonable predictions. An important feature of the semi-empirical incremental model is the definition of loading and unloading, which is different from that assumed in elasto-plasticity. This definition distinguishes between spherical and deviatoric loading/unloading. The definition of deviatoric loading/unloading has been subject to some criticism. It was therefore discussed and clarified in this paper on the basis of the experiment presented.
EN
Generalization of Linear Rosenstark Method of Feedback Amplifier Analysis to Nonlinear One This paper deals with an extension of the Rosenstark’s linear model of an amplifier to a nonlinear one for the purpose of performing nonlinear distortion analysis. Contrary to an approach using phasors, our method uses the Volterra series. Relying upon the linear model mentioned above, we define first a set of the so-called amplifier’s constitutive equations in an operator form. Then, we expand operators using the Volterra series truncated to the first three components. This leads to getting two representations in the time domain, called in-network and input-output type descriptions of an amplifier. Afterwards, both of these representations are transferred into the multi-frequency domains. Their usefulness in calculations of any nonlinear distortion measure as, for example, harmonic, intermodulation, and/or cross-modulation distortion is demonstrated. Moreover, we show that they allow a simple calculation of the so-called nonlinear transfer functions in any topology as, for example, of cascade and feedback structures and their combinations occurring in single-, two-, nd three-stage amplifiers. Examples of such calculations are given. Finally in this paper, we comment on usage of such notions as nonlinear signals, intermodulation nonlinearity, and on identification of transfer function poles and zeros lying on the frequency axis with related real-valued frequencies.
EN
The problem of dilation is discussed in the context of classical Cam-Clay model, which was developed on the basis of a specific assumption regarding the plastic work. This assumption leads to a special form of the dilation function, from which a shape of yield function is derived. The above mentioned assumption is compared with the results of the triaxial tests, performed on the model “Skarpa” sand. It is shown that the Cam-Clay approach is not realistic, as it is based on the assumption which is not consistent with experimental data. Some general considerations and discussion of this problem are also presented.
EN
The procedure of the formulation of constitutive equations for asphalt-aggregate mixes is based very often on rheological schemes composed of classical elastic, plastic and viscous elements. The parameters of these schemes can be obtained based on laboratory experiments. In order to obtain better curve fitting results one can use non-classical viscoelastic elements described by fractional derivatives. In this paper we present the characteristics of the fractional viscoelastic Huet-Sayegh model as well as the characteristics of an original simplified fractional model. The results have been obtained using algorithms of numerical calculation of inverse Laplace transforms. Then the proposal of an original rheological model including plasticity has been given. The non-linear differential constitutive relationships of such a model are presented in the paper. The results of computer simulations are also visualized. Finally, 3D viscoelasticplastic models of asphalt aggregatemixes are proposed. The models are based on a generalized macroscopic theory taking into account the effect of pressure-dependency on yielding.
EN
The paper deals with some constitutive equations for arterial walls subjected to uniaxial and biaxial extension tests. In spite of a great number of various approaches to development of the strain-stress relations, the models selected for presentation seems to be good representatives of the most characteristic and fundamental ones.
EN
In the work, research has been taken up on the problem of lead elimination from presently very popular technical Journal Composites [1] presented a paper by T. Niezgoda and M. Klasztorny, giving predicted values of elastic constants for a special case of balanced laminates. The present paper is influenced by the idea from the mentioned work, where the described case can be obtained without the use full methods of the theory of lamination, employing instead simple engineering techniques, which in turn can be solved symbolically and can bring a better understanding of the problems of lamination. The relations given here show the step by step procedure of obtaining the necessary elasticity relations and stress values. The main aim in presenting this paper is twofold. Primarily the employed methods are easy to understand and useful in the cases of first contact with the theory of lamination. Secondly, the method of obtaining presented results for diagonal-ply laminates, using a simple rotation of coordinates, is unique and to the Authors knowledge has not been published up to now. There are also other reasons for presenting this paper for not only there are insufficient data to follow the original procedure but also there is an error in the called literature, which now here is presented as [2]. Unfortunately this paper obtained by personal communication does not help much. Another item in this presentation is giving the exact, useful values of material constants for a diagonal laminate, together with values of acting stresses for this case, which may be of further use. As a result the present paper gives a detailed explanation of the procedure used in the homogenization process, shows application of this method to another commonly used composite and presents useful values of internal stresses, which decide on strength of the structure.
PL
Czasopismo Kompozyty [1] przedstawiło pracę T. Niezgody i M. Klasztornego, podającą przewidywane wartości stałych sprężystości specjalnego typu zrównoważonych laminatów. Praca obecna jest wywołana pomysłem pochodzącym z wymienionej publikacji, z którego wynika, że w specjalnych przypadkach możliwe jest przewidywanie własności laminatu, używając prostych metod inżynierskich, bez uciekania się do pełnej teorii laminowania. Przedstawienie niniejszej pracy ma dwa cele. Po pierwsze, zdaniem Autorów, zaprezentowana metoda jest bardzo przydatna, szczególnie w przypadku pierwszego kontaktu z teorią laminowania kompozytów i może być z pożytkiem stosowana podczas początkowych zajęć dydaktycznych z tych zagadnień. Przedstawiono krok po kroku proces tworzenia zależności opisujących wartości stałych sprężystości laminatu oraz wartości poszczególnych naprężeń. Celem drugim jest pokazanie wyjątkowo prostej metody uzyskiwania wyników w opisie laminatów diagonalnych, polegającej na prostym obrocie układu współrzędnych, nieznanej dotychczas zdaniem Autorów, z innych publikacji. Innym powodem opracowania obecnej pracy jest to, że ani z publikacji [1], ani z publikacji [2], błędnie cytowanej w wymienionej pracy przez obu autorów, a uzyskanej drogą osobistego przekazu, nie wynika jasno metoda postępowania, prowadząca do przedstawionych wyników. W efekcie tych stwierdzeń powstała niniejsza praca, pokazująca pełną drogę postępowania w badanych prostych przypadkach, co być może ułatwi przekazanie istoty aspektów teorii laminowania w prostej formie. Niezależnie od tego rozważono inny, równie prosty przypadek szczególny często występującego w praktyce laminatu, a także podano zależności umożliwiające określenie wartości naprężeń wewnętrznych, decydujących o wytrzymałości laminatu. Wreszcie w przypadkach prób oceny wpływu poszczególnych składników kompozytu czy laminatu na własności powstałego materiału zależności podane w formie symbolicznej okazują się bardzo przydatne.
14
Content available remote Multi-scale modelling of the mechanical behaviour of grouted sand
EN
Such mechanical properties of sand as its stiffness, cohesion and, to a less extent, friction angle can be increased through the process of grouting. A constitutive model adapted for cohesive-frictional materials from a homogenization technique, which allowed us to integrate constitutive relations on the grain level, has been developed to obtain constitutive equations for the equivalent continuous granular medium. The local behaviour was assumed to obey Hertz-Mindlin's elastic law and Mohr-Coulomb's plastic law. The influence of the cement grout was modelled by means of adhesive forces between grains in contact. The intensity of these adhesive forces is the function of nature and the amount of grout present inside the material and can be reduced due to a damage mechanism at the grain contact during loading. In this paper, we present several examples of simulation which show that the model can reproduce with sufficient accuracy the mechanical improvement induced by grouting as well as the damage of the grain cementation during loading.
EN
On the basis of modified Hooke’s law for multiaxial stress in viscoelastic solids, threedimensional constitutive equations for strains have been derived. It is shown that after application or removal of triaxial static load, normal and shear strain components vary in course of time proportionally to each other and that in-phase stress components produce in-phase strain components. Harmonic out-of-phase stress as well as multiaxial periodic and stationary random stresses are also considered. The matrix of dynamical flexibility of viscoelastic materials is determined which depends on three material constants (Young modulus, Poisson’s ratio and coefficient of viscous damping of normal strain) and load circular frequency.
EN
This paper presents an approach to describe the dynamic behaviour of a magnetorheological (MR) damper by the Bodner-Partom (B-P) constitutive equations. The B-P equations usually used for metal materials are presented for shear stresses to express viscoplastic proprieties of a MR fluid. The material parameters for the LORD RD 1005-3 damper are determined. The experimental results are shown and are compared with the numerical results.
17
Content available remote Studies on constitutive equation that models bone tissue
EN
A method for identifying viscoelastic constitutive equations for bone is developed. In the method, anisotropy, non-homogeneity and bone remodelling are taken into consideration. The equations correspond to monotropic rheological model of bone. In order to calculate the material parameters of elasticity and viscosity, a new algorithm is developed, in which the results of creep compression and shear tests are used. The way of determining the material constants of elasticity and viscoelasticity for bone in the areas that are crucial in strain and stress analysis is shown a well. The strength experiments (tests) and creep tests were performed on bone samples extracted from the femur of calf. The method of bone modelling in terms of rheology is the following: bone samples are properly prepared from biological material. For given bone samples, which have more or less the same density, three independent short-term creep tests are carried out. The results of the creep tests permit us to determine five elastic constants and viscoelastic constants of a monotropic material [1]. In order to describe rheological processes in terms of structural models, fractional exponential functions and normal exponential functions are used. The constitutive equations are formulated in the compliance form. The computer program that executes the algorithm of elastic and viscoelastic constant determination is used.
18
Content available remote On micro-damage in hot metal working. P. 2 Constitutive modelling
EN
Damage constitutive equations are formulated to model the evolution of grain boundary and plasticity-induced damage for free-cutting steels under hot forming conditions. During high temperature, high strain rate deformation, material degradation has characteristics of both creep damage at grain boundaries, and ductile damage surrounding hard inclusions. This has been experimentally observed and is reported in the companion paper. This paper describes the development of unified viscoplastic-damage constitutive equations, in which the nucleation and growth of both damage types are considered independently. The effects of deformation rate, temperature, and material microstructure on damage evolution are modelled. The proposed damage evolution equations are combined with a viscoplastic constitutive equation set, enabling the evolution of dislocation hardening, recovery, recrystallisation, grain size, and damage to be modelled. This set of unified, mechanism-based, viscoplastic damage constitutive equations is determined from experimental data of a freemachining steel for the temperature range 1173- 1373 K. The fitted model is then used to predict damage and failure features of the same material tested using a set of interrupted constant strain rate tests. Close agreement between the predicted and experimental results is obtained for all the cases studied.
19
Content available Wear patterns and laws of wear - A review
EN
Wear is a process of gradual removal of a material from surfaces of solids subject to contact and sliding. Damages of contact surfaces are results of wear. They can have various patterns (abrasion, fatigue, ploughing, corrugation, erosion and cavitation). The results of abrasive wear are identified as irreversible changes in body contours and as evolutions of gaps between contacting solids. The wear depth profile of a surface is a useful measure of the removed material. The definition of the gap between contacting bodies takes into account deformations of bodies and evolutions of wear profiles. The wear depth can be estimated with the aid of wear laws. Derived in this study, constitutive equations of anisotropic wear are extensions of the Archard law of wear. The equations describe abrasion of materials with microstructures. The illustrative example demonstrates calculations of the abraded mass and temperatures in pin-on-disc test rig.
PL
Zużycie jest to stopniowe usuwanie materiału z powierzchni stykających i ślizgających się ciał stałych. Następstwem zużycia są uszkodzenia powierzchni styku. Mogą one przyjmować różną formę (ścierania, zmęczenia, bruzdowania, korugacji, erozji i kawitacji ) Konsekwencjami procesu zużycia ściernego są nieodwracalne zmiany kształtów ciał oraz powiększanie luzów (szczelin) między stykającymi się ciałami. Użyteczną miarą usuniętego materiału jest profil głębokości zużycia powierzchni. Definicja luzu między stykającymi się ciałami uwzględnia odkształcenia ciał oraz ewolucję profili zużycia. Głębokość zużycia może być oszacowana z pomocą praw zużycia. Wyprowadzone w niniejszej pracy równania konstytutywne anizotropowego zużycia są rozszerzeniem prawa zużycia Archarda. Opisują one ścieranie materiałów z mikrostrukturą. W przykładzie ilustracyjnym obliczono ubytek masy i temperaturę w stanowisku doświadczalnym typu pin-on-disc.
20
Content available remote Badania reologiczne i modelowanie konstytutywne żywicy winyloestrowej Atlac 590
PL
Opracowano model reologiczny i sformułowano standardowe sprzężone równania konstytutywne lepkosprężystości żywicy winyloestrowej Atlac 590, z uwzględnieniem liniowej niezależności odkształceń kierunkowych. Zastosowano tylko funkcje wykładnicze ułamkowe jako funkcje tworzące. Wyznaczono stałe lepkosprężystości z próby pełzania klasycznego, trwającej 90 dni i zbadano wpływ fluktuacji temperatury otoczenia.
EN
A rheological model for a plastic made of Atlac 590 vinylester resin has been developed. Standard coupled constitutive eąuations of yiscoelasticity describing this model have been formulated under assumption that the directional strains arę linearly independent. There are only applied fractional exponential functions as the generating functions. The viscoelastic constants have been derived from the classic creep experiment lasting 90 days. The influence of the environment temperature fluctuations has been checked.
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