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This paper provides a thorough investigation of a heat conduction problem that pertains to tolerance modelling in layered materials made up of multiple components. These media are functionally graded materials and thus have varying properties that affect their effectiveness. The proposed equations explain the conduction of heat in layered composites. The formulation involves partial differential equations, which utilise smooth and slowly varying functions. Notably, an extension of the unified tolerance modelling procedure is presented generalising existing models for two-component step-wise functionally graded materials (FGMs). This extension allows for the analysis of specific issues related to heat conduction in multi-component stratified composites with a transversal gradation of effective properties. This is the most important novelty achievement of the present paper because it will contribute to advancing knowledge and allows researchers, engineers, and practitioners to use the method in a broader context, addressing a more extensive set of real-world situations not limited to the number of component materials. [ABSTRACT FROM AUTHOR]

Dynamic problems of elastic non-periodically laminated solids are considered in this paper. It is assumed that these laminates have a functionally graded structure on the macrolevel along the x1-axis and non-periodic structure on the microlevel. However, along the other two directions, i.e., x2 and x3, their properties are constant. The effects of the size of a microstructure (the microstructure effect) on the behaviour of the composites can play a significant role. This effect can be described using the tolerance modelling method. This method allows us to derive model equations with slowly varying coefficients. Some of these terms can depend on the size of the microstructure. These governing equations of the tolerance model make it possible to determine formulas describing not only fundamental lower-order vibrations related to the macrostructure of these composite solids, but also higher-order vibrations related to the microstructure. Here, the application of the tolerance modelling procedure is shown to lead to equations of the tolerance model that can be used for non-periodically laminated solids. Then, these model equations are mainly used to analyse a simple example of vibrations for functionally graded composites with non-periodically laminated microstructure (FGL). Similar problems were investigated in the framework of the homogenised (macrostructural) model (Jędrysiak et al. 2006); the resulting equations neglect the microstructure effect. [ABSTRACT FROM AUTHOR]

Some considerations of slender elastic nonperiodic beams are shown in this paper. These beams have a functionally graded structure on the macro-level along the x-axis, and a nonperiodic structure on the micro-level. The effect of the size of the microstructure on the behavior of the beams can play a crucial role. This effect can be taken into account by applying the tolerance modelling method. This method leads to model equations with slowly varying coefficients, some of which depend on the microstructure size. In the framework of this model, formulas of higher order vibration frequencies related to the microstructure can be determined, not only for the fundamental lower-order vibration frequencies. Here, the application of the tolerance modelling method was mainly shown to derive the model equations of the so-called general (extended) tolerance model and standard tolerance model, describing dynamics and stability for axially functionally graded beams with the microstructure. A simple example of free vibrations of such a beam was presented as an application of these models. The formulas of the frequencies were determined using the Ritz method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Thin elastic periodic plates are considered in this paper. Since the plates have a microstructure, the effect of its size on behaviour of the plates can play a crucial role. To take into account this effect, the tolerance modelling method is applied. This method allows us to obtain model equations with constant coefficients, which involve terms dependent of the microstructure size. Using the model equations, not only can formulas of fundamental lower-order vibration frequencies be obtained, but also formulas of higher-order vibration frequencies related to the microstructure. In this paper, the effect of the material periodic microstructure on free vibration frequencies for various boundary conditions of the plates was analysed. To obtain proper formulas of frequencies, the Ritz method is applied. Moreover, some results are compared to the results calculated using the FEM. [ABSTRACT FROM AUTHOR]

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The object of the analysis is a two-dimensional elastostatic problem for multicomponent, multilayered periodic composites. The equations of equilibrium for this composite are obtained within the framework of tolerance modelling procedure. The paper presents two examples of solutions of boundary value problems.

Abstract The subject of this article is analysis of transverse vibrations of beams which geometric and material properties vary periodically along the longitudinal axis. The aim is to present averaged models that take into account the shear deformation and geometric non-linearity, and to analyse transverse vibrations of such beams in moderately large deflection range. As the theoretical foundations, we use Timoshenko beam theory with von Kármán-type non-linearity. This results in obtaining new differential equations with constant coefficients, some of which explicitly depend on the beam inhomogeneity period size. Then, a reasonably simplified model is proposed to describe the vibrations of the considered beams in the low frequency range. The differential equations are transformed into a system of algebraic equations according to the Galerkin method. The response of the beam to transverse harmonic load is investigated by means of a pseudo arc-length continuation scheme. Non-linear coupling between vibration modes and the possibility of superharmonic resonance occurrence are taken into account. As an example of application, few special cases of beam geometry and boundary conditions are examined and compared. The results have the potential application to structural vibration control. [ABSTRACT FROM AUTHOR]

The present study aims to analyse a two-dimensional problem of displacements in theory of thermal stresses for multicomponent, multi-layered periodic composites. The model equations are obtained within the framework of the tolerance modelling procedure. These equations allow to determine the distribution of displacements caused by the temperature field in the theory of thermal stresses. The paper presents an example of a solution of a boundary value problem. [ABSTRACT FROM AUTHOR]

The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperature field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM-type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Boundary effect behavior understood as near-boundary suppression of boundary fluctuation loads is described in various ways depending on the mathematical representation of solutions and the type of the center. In the case of periodic composites, the homogenization method is decisive here. In the framework of the Tolerance Averaging Approach, developed by prof. Cz. Woźniak leading to an approximate model of phenomena related to periodic composites this effect is described by a homogeneous part of differential equation for fluctuation amplitudes and usually this approximate description of the boundary effect behavior is restricted to a single fluctuation. In this paper, contrary to the previous elaborations, the boundary effect is developed in the variant of the tolerance thermal conductivity model in which the temperature field is represented by the Fourier expansions composed by an average temperature with infinite number of Fourier terms imposed on the average temperature as tolerance fluctuation suppressed in the framework of the boundary effect. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

A tolerance modelling approach for problems of periodic media and structures is adopted here to investigate free vibrations of thin functionally graded plates with a microstructure along one direction in the planes parallel to the plate midplane. The size of the microstructure is assumed to be of an order of the plate thickness. To show the effect of the microstructure on dynamic behaviour of these plates the tolerance modelling method is used, cf. Jędrysiak (2013). This method leads to model equations with smooth functional coefficients, which describe the effect of the microstructure size of these plates not only in dynamic but also in stationary problems. In order to take into account this effect two models – the asymptotic-tolerance model and the tolerance model are used. As an example of applications of these models free vibration frequencies for simply supported rectangular plate are calculated. A certain verification for some cases is also shown by using the finite element method. [ABSTRACT FROM AUTHOR]

The objects of consideration are thin linearly thermo-elastic Kirchhoff-Love-type circular cylindrical shells having a periodically micro-inhomogeneous structure in circumferential direction (uniperiodic shells). The aim of this note is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic thermoelastic problems for these shells. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, the proposed tolerance model equations have constant coefficients depending also on a cell size. Hence, an important advantage of this model is that it makes it possible to investigate the effect of a period of inhomogeneity on the global shell thermodynamics (the length-scale effect). This effect is neglected in the known homogenized models derived by asymptotic methods. [ABSTRACT FROM AUTHOR]

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 aim of this note is to formulate and discuss a new non-asymptotic averaged model for the analysis of selected dynamic problems for these shells. The proposed asymptotic-tolerance model equations have continuous and slowly varying coefficients depending also on a cell size. An important advantage of this model is that it makes it possible to study micro-dynamics of tolerance-periodic shells independently of their macro-dynamics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this note the problem of thermoelasticity in composite, made of two components non-periodically distributed as microlaminas along one direction x 1 , which macroscopic properties change continuously along this direction perpendicular to the laminas, is considered. Two models are presented: the tolerance model, which governing equations take into account the effect of the microstructure size on the overall behaviour of these laminates and the asymptotic model, in which equations this effect is omitted. The tolerance model equations are obtained using the tolerance averaging technique. Applications of both models are shown in some examples. Then the results obtained by the tolerance modelling are verified with the exact solution. [ABSTRACT FROM AUTHOR]

The object of analysis is a heat conduction problem within the framework of tolerance modelling in laminated media with a functional gradation of effective properties. In contrast to the known asymptotic models (based on the homogenization technique), the characteristic feature of the tolerance model equations is that they make it possible to analyze the effect of the microstructure size on the overall behavior of the laminate. The proposed model equations describe heat conduction in laminates by mean of partial differential equations with smooth and slowly varying functions. This paper describes a certain extension of the unified tolerance modelling procedure, which makes it possible to analyze specific problems of heat conduction in laminates with a transversal gradation of effective properties. [ABSTRACT FROM AUTHOR]

There are considered thin periodic plates with moderately large deflections. To take into account the effect of the microstructure on behaviour of these plates the tolerance modelling method is applied, cf. Domagalski and Jędrysiak [2] , Meccanica, 2012. This method makes it possible to derive model equations with constant coefficients involving terms dependent of the microstructure size. The paper contains an computational example of critical load calculations and postbuckling analysis. [ABSTRACT FROM AUTHOR]

There are considered functionally graded plates with a microstructure. The size of the microstructure is of an order of the plate thickness. It is assumed that the functionally graded macrostructure of these plates is only along one axis. Because the effect of the microstructure on dynamic behaviour of the plates under consideration plays a crucial role the tolerance modelling method is applied, cf. Kaźmierczak and Jędrysiak, Engng. Struct., 2013, cf. Jędrysiak, Thin-Walled Struct., 2013. This method makes it possible to obtain model equations with smooth functional coefficients, describing the effect of the microstructure size. In this paper it is shown that this effect can be taken into account in two models: the asymptotic-tolerance model and the tolerance model. Applications of these models with a comparison to results by the asymptotic model are presented for various boundary conditions and different distribution functions of material properties in an analysis of free vibration frequencies of functionally plate strips. Moreover, a certain verification for chosen cases is also shown by using the finite element method. [ABSTRACT FROM AUTHOR]

Non-nominal visualisation is used to evaluate how geometrical variation influences the appearance of split-lines between components on a car body. Depending on the simulation method, visualised variation can be represented in an unrealistic manner, which can affect the perception of the result. This paper presents a comparative study in which eye-tracking equipment has been used in order to explore whether the interpretation of variation differs between rigid- and non-rigid-based variation simulations during the evaluation of visualised models. Subjects from the automotive industry were asked to evaluate two virtual models where the same amount of variation was represented by rigid and non-rigid models. Eye tracking was used to record how the subjects performed the assessment. The result shows that there is a significant difference in how simulation results are interpreted using rigid versus non-rigid simulation. [ABSTRACT FROM PUBLISHER]

Abstract: Functionally graded plates with a special microstructure are considered. It is assumed that the size of the microstructure is of an order of the plate thickness. Moreover, these plates have functionally graded macrostructure along only one axis. To take into account the effect of the microstructure on dynamic behaviour of the plates under consideration the tolerance modelling method is applied, [cf. Kaźmierczak, Jędrysiak, Thin-Walled Structures 2011] Using this method there are obtained model equations with smooth functional coefficients involving terms dependent of the microstructure size. As an example free vibration frequencies of a plate band are investigated. [Copyright &y& Elsevier]

Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applications, these models are used to analyse a bending of periodic plates under various loadings. [ABSTRACT FROM AUTHOR]

Abstract: A problem of a modelling of vibrations of thin plates with a functionally graded macrostructure and a tolerance-periodic microstructure in planes parallel to the plate midplane is analysed. The proposed model, based on the Kirchhoff plate theory assumptions and additional hypothesis of the tolerance averaging technique [Woźniak et al., editors. Thermomechanics of heterogeneous solids and structures. Tolerance averaging approach. łódź, University Press, Technical University of łódź; 2008], describes the effect of the microstructure size on dynamic behaviour of the plate. In this paper, as an example there are analysed free vibration frequencies of a functionally graded plate band. These frequencies are obtained in the framework of two proposed models—the tolerance model and the asymptotic model, using the known Ritz method. [Copyright &y& Elsevier]

In this contribution a problem of heat conduction in laminates, made of two conductors nonperiodically distributed as microlaminas along one direction, is investigated. A macrostructure of such laminate is assumed to be functionally graded along this direction. In order to analyse some effects of the microstructure on the heat conduction, the tolerance averaging technique is applied, cf. the book edited by Woźniak, Michalak and Jędrysiak [1]. [ABSTRACT FROM AUTHOR]

The object of considerations is a multilayered composite, made of two components non-periodically distributed as microlaminas along one direction. Macroscopic properties of this composite are assumed to be continuously varied across laminas. The aim of this note is to propose two models of thermoelasticity problems: the tolerance model, taking into account the effect of the microstructure size on the overall behaviour of these composites, and the asymptotic model, neglecting this effect. The governing equations of both the models are derived by using averaging procedures presented in the book edited by Cz. Woźniak et al. [1], i.e. the tolerance modelling and the consistent asymptotic modelling. [ABSTRACT FROM AUTHOR]

This article presents a tolerance modelling and robust design approach to support concurrent engineering. This method allows the designer to synthetically specify dimensional and geometric tolerance, considering assembly functional requirements (AFRs) and manufacturing costs. First, features of ISO/TC 213 are used as the basis for the construction of tolerance network and tolerance model for assembly. Second, the manufacturing cost-tolerance model for cylindrical and planar features is established. This model addresses not only dimensional tolerances but also geometric tolerances and nominal parameters. Finally, the robust tolerance optimization model is established, and genetic arithmetic is used to obtain robust tolerance values. The proposed approach is consistent with the philosophy of concurrent engineering, in which AFRs are satisfied and manufacturing cost is reduced. A design instance is introduced to show the validity of this method. [ABSTRACT FROM AUTHOR]

An optimal approach for process tolerance balancing is presented. The new approach is based on process capabilities and is to be used in the stage of process planning. A nonlinear programming model is used to simultaneously optimise process tolerances of required operations. In the optimisation model, the objective function is to minimise the total manufacturing cost with different weighting factors. Using the estimated standard deviations of the dimensions and the manufacturing cost-tolerance functions, the constraint equations for the process tolerance chains and the manufacturing capability indices are established, together with a model for the economical tolerance bounds of machine tools. A practical example was used to verify the usefulness of the proposed approach. The results of the comparative study show that the proposed approach is more advantageous in relaxing tolerance requirements and in reducing scrap rates generally, compared with the existing methods. [ABSTRACT FROM AUTHOR]

This paper deals with the linear natural vibrations analysis of beams where the geometric and material properties vary periodically along the beam axis. In contrast with homogeneous prismatic beams, the frequency spectra of such beams are irregular as there exist enlarged intervals between some adjacent frequencies. Presented here are two averaged models of beams based on the tolerance modelling approach. The assumptions of classical Euler–Bernoulli and Timoshenko–Ehrenfest beam theories are adopted as the foundations. The resulting mathematical models are systems of differential equations with constant, weight-averaged coefficients. This makes it possible to apply any classical method of solution suitable for homogeneous beams, such as Galerkin orthogonalization. Here, emphasis is placed on the comparison of natural frequencies neighbouring the frequency band-gaps that are obtained from these two theories. Two basic cases of material and geometric property distribution in a periodicity cell are studied, and the natural frequencies and mode shapes are obtained for a simply supported beam. The results are supported by a comparison with the finite element method and partially exact solutions. [ABSTRACT FROM AUTHOR]

• The vibration analysis of sandwich microheterogeneous plates is performed. • A set of assumptions of a versatile and sufficiently precise model is determined. • The analysed models must be convenient for use for engineers. • The tolerance averaging technique is used within the modelling procedure. 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. [ABSTRACT FROM AUTHOR]

In this paper mathematical modelling of vibrations and stability problems for slender visco-elastic periodic beams is considered. In order to take into account the effect of microstructure a certain non-asymptotic approach is applied, called the tolerance modelling method. This technique allows to replace the equation with non-continuous, highly oscillating, periodic coefficients by a system of differential equations with constant coefficients. Moreover, the derived equations describe the effect of microstructure on the overall behaviour of the beams under consideration. In the framework of the tolerance modelling governing equations of two different tolerance models can be obtained – the standard and the general, under weakened assumptions. To evaluate the proposed general tolerance model obtained results are compared with results calculated within the known asymptotic homogenization, i.e. by the asymptotic model. In this note mathematical models describing investigated beams are only presented and illustrated by a simple example. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The mathematical modelling of certain problems of vibrations and stability for periodic slender visco-elastic beams is presented in this note. To consider these problems and take into account the effect of the microstructure, the tolerance modelling approach is proposed. Using this technique, the equation with non-continuous, periodic, highly oscillating coefficients is replaced by a system of differential equations with constant coefficients. Moreover, these governing equations describe the effect of the microstructure on the overall behavior of the beams under consideration. The tolerance modelling can lead to equations of two different tolerance models—the standard and the general, under weakened assumptions. This averaging tolerance method was assessed by comparison with the asymptotic homogenization, the governing equations of which omit this effect. My considerations were limited to proposing and presenting only mathematical models describing investigated beams. In a simple analytical example, the application of the presented average models is shown. [ABSTRACT FROM AUTHOR]