Tolerance Averaging Technique Research Articles

Mathematical modelling of stability problems for thin transversally graded cylindrical shells

The objects of consideration are thin linearly elastic Kirchhoff–Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new mathematical averaged non-asymptotic model for the analysis of selected stability problems for such shells. As a tool of modelling we shall apply the tolerance averaging technique. The second aim is to derive and discuss a new mathematical averaged asymptotic model. This model will be formulated using the consistent asymptotic modelling procedure. The starting equations are the well-known governing equations of linear Kirchhoff–Love second-order theory of thin elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in circumferential direction, whereas equations of the proposed models have continuous and slowly-varying coefficients. Moreover, some of coefficients of the tolerance model equations depend on a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental cell-independent, but also the new additional cell-dependent critical forces can be derived and analysed.

Influence of Composite Structure on Temperature Distribution-An Analysis Using the Finite Difference Method.

Among composites, we can distinguish periodic structures, biperiodic structures, and structures with a functional gradation of material properties made of two or more materials. The selection of the composite's constituent materials and the way they are distributed affects the weight of the composite, its strength, and other properties, as well as the way it conducts heat. This work is about studying the temperature distribution in composites, depending on the type of component material and its location. For this purpose, the Tolerance Averaging Technique and the Finite Difference Method were used. Differential equations describing heat conduction phenomena were obtained using the Tolerance Averaging Technique, while the Finite Difference Method was used to solve them. In terms of results, temperature distribution plots were produced showing the effect of the structure of the composite on the heat transfer properties.

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core.

Sandwich structures are commonly used in many branches of modern engineering, such as aerospace or naval constructions. In this work, a vibration analysis of such structures is performed with the use of an anlytical model based on a zig-zag hypothesis. Due to the assumed periodic microstructure, which may occure in any layer of the structure, the initial governing equations describing its dynamic behaviour may contain periodic, non-continuous coefficients. The main aim of the presented paper is to show an analytical solution to the issue of the vibration analysis of the mentioned structures. With the use of the tolerance averaging technique, the initial governing equations are transformed to the form with constant coefficients, which is convenient to solve using well-known mathematical methods. The derived model is a versatile solution for any type of periodically inhomogeneous sandwich plate, including sandwich plates with a honeycomb core. Eventually, in the calculation example, the application of the derived averaged model in the analysis of vibrations of such structures is presented and discussed. The convergence of results of the tolerance model and FEM analysis proves the correctness and superiority of the proposed solution.

Mathematical modelling of thermoelasticity problems for thin biperiodic cylindrical shells

The objects of consideration are thin linearly thermoelastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to formulate and discuss two new averaged mathematical models for the analysis of selected dynamic thermoelasticity problems for the shells under consideration: the non-asymptotictolerance and the consistent asymptotic models. The starting equations are the well-known governing equations of linear Kirchhoff-Love theory of thin elastic cylindrical shells combined with Duhamel–Neumann thermoelastic constitutive relations and coupled with the known linearized Fourier heat conduction equation in which the heat sources are neglected. For the microperiodic shells under consideration, the starting equations mentioned above have highly oscillating, non-continuous and periodic coefficients. The tolerance model is derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. It has constant coefficients depending also on a cell size. Hence, this model makes it possible to study the effect of a microstructure size on the global shell thermoelasticity (the length-scale effect). The consistent asymptotic model is obtained using the consistent asymptotic approach. It has constant coefficients being independent of the period lengths. Moreover, the comparison between the tolerance model for biperiodic shells proposed here and the known tolerance model for cylindrical shells with a periodic structure in the circumferential direction only (uniperiodic shells) is presented.

The Stability Analysis of Periodic Beams Interacting with Periodic Elastic Foundation with the Use of the Tolerance Averaging Technique.

In this paper a stability analysis of microperiodic beams resting on the periodic inhomogeneous foundation is carried out. The main issue of this considerations, which is the analytical solution to the governing equations characterised by periodic, highly oscillating and non-continuous coefficients, is overwhelmed by the application of the tolerance averaging technique. As a result of such application, the governing equation is transformed into a form with constant coefficients which can be solved using well-known mathematical methods. In several calculation examples, the convergence of the results of two derived averaged models is examined, as well as the convergence of the lowest value of the critical force parameter derived from the averaged models with the FEM model. The results prove the superiority of the presented analytical solution over the FEM analysis in the optimisation process.

Heat Transfer in Periodically Laminated Structures – Third Type Boundary Conditions

This work is about a heat transfer phenomenon in relation to the periodically laminated composite. The specific type of thermal loading, analyzed in this paper, require formulation of Robin boundary conditions. To consider a layered structure of analyzed composite, the tolerance averaging technique is used. This method allows to take into account a thickness of the layers and obtain the equations with continuous coefficients. To solve these equations, the finite difference method is used, because an analytical solution is not available in this case (in contrast to the analogous issue in relation to a homogeneous layer).

The tolerance modelling of vibrations of periodic sandwich structures – Comparison of simple modelling approaches

In literature one can find many different modelling approaches towards sandwich structures, such as: double-plate system with an elastic core, multi-layered plate, models based on advanced deformation hypothesis, etc. Those models are significantly varying from each other with a set of assumptions made at the beginning of the modelling process. In order to determine the set of assumptions of the simple and sufficiently precise model of the sandwich plate, which would be convenient for engineers, the assumptions made in four basic models of sandwich plates are presented and discussed. Consequently, the initial systems of governing equations for vibration analysis of sandwich structures made of isotropic materials are derived. Additionally, those equations are transformed with the use of the tolerance averaging technique, so as the structures with a periodic microstructure can be also analysed. Eventually, the application of the derived models in the free vibration analysis is presented and used to assess the correctness of the modelling assumptions of every each model.

A comparison of dynamic models of microheterogeneous asymmetric sandwich plates

Due to a lack of analytical solution to the issue of dynamics of asymmetric sandwich plates with a periodic microstructure, the only available method of modelling of such structures is FEM. However, such approach is usually a time-consuming process, which additionally requires a lot of computing resources due to a highly refined mesh. In this paper the analytical model of the mentioned structure, based on the tolerance averaging technique, is presented and discussed. The obtained model can be formulated with a system of partial differential equations with constant coefficients, which is relatively simple to solve. As a result, the proposed method is convenient for engineers and accurate. Moreover, unlike the asymptotic homogenisation method, it can be used to investigate the influence of microstructure on the overall behaviour of the plate. Eventually, in the calculation example a comparative simulations were conducted to investigate the influence of certain set of assumptions on the obtained results of free vibration frequencies and to verify the effectiveness and superiority of proposed calculation method over the FEM.

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation.

In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces.

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

On the Effect of Material Properties Uncertainty on the Magnitude of Temperature Oscillations in Two-Component Periodic Laminate: Tolerance Averaging Modelling

Over last decades composite materials gained even more interests in many industries due to theirs effective properties which may be apparently different (better) then constituents itself. By specific layout and distribution of composite components we can achieve desired properties in macro scale, e.g. high elasticity and low conductivity at the same time. On all interfaces, by perfect contact between phases, there will appear jump of gradient field (displacement or temperature) of unknown magnitude. This magnitude depends in fact, for the heat transfer problem, inter alia on the ratio of conductivities of composite constituents lying next to interface. Values of these oscillation magnitudes are case of our study here. The conductor under considerations is a two-phase, periodic laminate subjected to initial-boundary conditions assuring unidirectional heat flow, perpendicular to the laminas. Ratios of material properties are assumed as random variables of known probabilistic distribution. We will give an answer to the question: is the jump function of temperature gradient a random variable of Gaussian distribution In order to have a good description to considered problem we have decided on the use of tolerance averaging technique.

Free vibrations spectrum of periodically inhomogeneous Rayleigh beams using the tolerance averaging technique

In this paper, linear-elastic Rayleigh beams with a periodic structure are considered. Dynamics of such beams is described by partial differential equations with non-continuous highly oscillating coefficients. The analysis of dynamic problems using the aforementioned equations is very often problematic to perform. Thus, other simplified models of Rayleigh beams are proposed. Some of these models are based on the concept of the effective stiffness. Among them, one can distinguish the theory of asymptotic homogenization. However, in these models, the size of the mesostructure parameter (the size of a periodicity cell) is often neglected. Therefore, a non-asymptotic averaged model of the periodic beam is introduced, called the tolerance model, which is derived by applying the tolerance averaging technique (TA). The obtained tolerance model equations have constant coefficients, and in contrast to other averaged models, some of them depend on the size of the periodicity cell.

On Thermoelasticity in FGL – Tolerance Averaging Technique

Abstract In this paper the problem of linear thermoelasticity in a laminate with functional gradation of properties is considered. In micro level this laminate is made of two different materials, microlaminas, distributed non-periodically but also not randomly along one of directions, what in macro level results in aforementioned functionally gradation of laminate properties. In order to describe behavior of such structure, equations of two models are here presented – the tolerance and the tolerance-asymptotic model. Both are obtained by the tolerance averaging technique. The basic aim of this work is to analyse the influence of some terms from these averaged equations on the distribution and the values of the displacements and the temperature functions. To solve the equations of two proposed models the finite difference method is used.

On the Modeling of Periodic Sandwich Structures with the Use of the Broken Line Hypothesis

Abstract In this paper a dynamic analysis of sandwich plate with a certain periodic microstructure is considered. The initial system of governing equations is derived basing on the classic broken line hypothesis. As a result of transformations one can obtain a system of three differential equations of motion with periodic, highly oscillating and non-continuous coefficients. In order to derive a system of equations with constant coefficients tolerance averaging technique is applied. Eventually, in the calculation example a free vibration analysis of certain periodic plate strip is performed with the use of both the derived model and a FEM model. It can be observed that the consistency of obtained results is highly dependent on the calculation assumptions.

Some remarks on modelling of vibrations of periodic sandwich structures with inert core

In literature one can find many different modelling approaches towards sandwich structures, such as: double-plate system with an elastic core, multi-layered plate, models based on advanced deformation hypothesis, etc. Those models are significantly varying from each other with a set of assumptions made at the beginning of the modelling process. In order to determine the set of assumptions of the simple and sufficiently precise model of the sandwich plate, which would be convenient for engineers, the assumptions made in four basic models of sandwich plates are presented and discussed. Consequently, the initial systems of governing equations for vibration analysis of sandwich structures made of isotropic materials are derived. Additionally, those equations are transformed with the use of the tolerance averaging technique, so as the structures with a periodic microstructure can be also analysed. Eventually, the application of the derived models in the free vibration analysis is presented and used to assess the correctness of the modelling assumptions of every each model.

Effect of microstructure in thermoelasticity problems of functionally graded laminates

In this work the problem of thermoelasticity in composite, made of two different materials non-periodically distributed as microlaminas along one direction, is considered. The macroscopic properties of this composite are changing continuously along direction perpendicular to the laminas. In this note three models are presented: the tolerance and the asymptotic-tolerance model, taking into account the effect of the microstructure size and the asymptotic model, which equations omit this effect. To solve the equations of these three models, obtained by using the tolerance averaging technique, the finite difference method was used. Then the results obtained by the tolerance modelling were verified with the finite element method.

Tolerance and asymptotic modelling of dynamic problems for thin microstructured transversally graded shells

The objects of consideration are thin linearly elastic Kirchhoff-Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The first aim of this contribution is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic problems for these shells. As a tool of modelling we shall apply the tolerance averaging technique. Contrary to the starting exact shell equations with highly oscillating, non-continuous and tolerance-periodic coefficients, governing equations of the proposed tolerance model have continuous and slowly varying coefficients depending also on a cell size. Hence, an important advantage of this model is that it makes it possible to study the effect of a microstructure size on the global shell dynamics (the length-scale effect). The second aim is to derive and discuss a certain asymptotic model being independent of a microstructure size. It will be shown that in the framework of the tolerance model not only the fundamental lower, but also the new additional higher-order free vibration frequencies can be derived and analysed.

Heat conduction in periodic laminates with probabilistic distribution of material properties

This contribution deals with a problem of heat conduction in a two-phase laminate made of periodically distributed micro-laminas along one direction. In general, the Fourier’s Law describing the heat conduction in a considered composite has highly oscillating and discontinuous coefficients. Therefore, the tolerance averaging technique (cf. Woźniak et al. in Thermomechanics of microheterogeneous solids and structures. Monografie - Politechnika Łodzka, Wydawnictwo Politechniki Łodzkiej, Łodź, 2008) is applied. Based on this technique, the averaged differential equations for a tolerance–asymptotic model are derived and solved analytically for given initial-boundary conditions. The second part of this contribution is an investigation of the effect of material properties ratio \(\omega\) of two components on the total temperature field \(\theta\), by the assumption that conductivities of micro-laminas are not necessary uniquely described. Numerical experiments (Monte Carlo simulation) are executed under assumption that \(\omega\) is a random variable with a fixed probability distribution. At the end, based on the obtained results, a crucial hypothesis is formulated.

A contribution to the modelling of heat conduction for cylindrical composite conductors with non-uniform distribution of constituents

This contribution deals with a heat conduction in a certain composite conductor with a deterministic structure, which is not periodic but in small regions of a conductor can be approximately regarded as periodic. The aim of this contribution is to formulate and apply two mathematical models for the analysis of heat transfer in skeletal rigid conductors. The formulation of the macroscopic mathematical models for analysis of heat transfer in these conductors will be based on the tolerance averaging technique (cf. monographs edited by Woźniak et al. (2008, 2010)). The general results are applied to the analysis of some special problems. It will be carried out validation of the obtained mathematical model by comparison of results from obtained tolerance-asymptotic model equations with results from finite elements method (Abaqus program).

Tolerance modelling of vibrations of periodic three-layered plates with inert core

In this note a free vibration analysis of periodic three-layered sandwich structures is performed. Basing on the Kirchhoff’s thin plate theory simplified equations of motion are derived, which are characterised by highly-oscillating, periodic and non-continuous coefficients. In order to obtain a system of equations with constant coefficients, the tolerance averaging technique is used. An application of the proposed tolerance model to analyse free vibration frequencies of a three-layered plate strip is shown – for both lower order frequencies related to its macrostructure and higher order frequencies related to its microstructure. Some comparisons of results of lower frequencies, obtained in the tolerance, the asymptotic and the known homogenised models are presented. Moreover, a certain verification of the proposed model is performed using the Ritz method. It can be observed that the tolerance model can be successfully applied to analyse vibration problems of vast variety of periodic three-layered plates and can significantly improve the optimisation process of such structures.