تجزیه و تحلیل حساسیت از ویژگی های همگن برای بعضی از مواد مرکب الاستیک

The main goal of the paper is to present theoretical aspects and the finite element method implementation of sensitivity analysis in homogenization of composite materials with linear elastic components using the effective modules approach. The sensitivity analysis of effective material properties is presented in a general form for n-component periodic composite and is illustrated using the examples of periodic 1D as well as 2D heterogeneous structures. The sensitivity coefficients are determined for the effective Young’s modulus and the effective elasticity tensor components. The structural response functional for the fiber-reinforced elastic composite is proposed in the form of total strain energy resulting from some uniform strain state of the composite representative volume element (RVE). The results of sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials and its application in composite optimization, identification, shape sensitivity studies and, after some probabilistic extensions, in stochastic analysis of random composites.

As it is known, the sensitivity analysis in engineering systems is employed to verify how input parameters of a specific engineering problem influence the state functions (displacements, stresses, temperatures, for instance) [7], [8], [17] and [19]. The sensitivity coefficients, being the purpose of such an analysis, are computed using partial derivatives of the considered state function with respect to a particular input parameter. These derivatives can be obtained starting from fundamental algebraic equations system of the problem, for instance or, alternatively, by a simple derivation if a closed form solution exists. It is important to underline that this methodology is common for all discrete numerical techniques: boundary element method (BEM), finite difference method (FDM), finite element method (FEM) as well as hybrid and meshless strategies [16]. From the computational point of view, there are the following numerical methods in structural design sensitivity analysis [5], [8], [17] and [19]: the direct differentiation method (DDM), the adjoint variable method (AVM) applied together with the material derivative approach (MDA) or the domain parametrization approach (DPA) suitable for shape sensitivity studies [4]. Considering these capabilities and, on the other hand, a very complex structure of composite materials, sensitivity analysis should be applied especially in design studies for such structures. Instead of a single (or two) parameters characterizing elastic response of homogeneous structure, the total number of design parameters is obtained as a product of components number in a composite and the number of material and geometrical parameters for a single component. Some extra state variables should be analyzed to define interfacial behavior, general interaction of the constituents and/or the lack of periodicity. Usually, to reduce the complexity of original composite, so-called effective homogenization medium having the same strain (or complementary) energy is analyzed. This paper is devoted to general computational sensitivity studies of the homogenization method for some periodic composite materials with linear elastic and transversely isotropic constituents. The composite is first homogenized––the effective material tensor components are computed using the FEM-based additional computer program; material parameters of the composite most decisive for its effective material properties are determined numerically. It should be underlined that the homogenization method is generally an intermediate numerical tool applied to exclude the necessity of composite micro-scale discretization and, in the same time, to reduce the total number of degrees of freedom of the entire model. On the other hand, quite various numerical homogenization techniques are observed. They can be divided generally into two essentially different approaches: stress averaging (the boundary stresses are introduced between the composite constituents plus displacement-type periodicity conditions) [11], [12], [13], [14] and [15] and strain approach (uniform extensions of the RVE boundaries in various directions plus periodicity conditions on the remaining cell edges) [6]. Considering this, different results of the homogenization method in terms of the effective material tensors are obtained and hence quite different sensitivity gradients must be computed in these two approaches. The sensitivity analysis introduces a new aspect of the homogenization technique––it can be verified if the homogenized and original structures have the same or even analogous (in terms of their signs) sensitivity gradients. Then the composites can be optimized by manipulation with its material parameters or by selection of various constituent materials with computationally determined shape to the new designed composite structure. The sensitivity gradients are computed here by the use of homogenization-oriented computer program MCCEFF [12], [13], [14] and [15] according to the DDM approach implementation and presented as functions of the composite design parameters––Young’s moduli and Poisson’s ratios of the constituents. Since a finite difference scheme is used for the sensitivity gradients computations, numerical sensitivity of final results to the increment of arbitrarily introduced parameter must be verified. This numerical phenomenon makes it necessary to determine the most suitable interval of parameters increments for the particular effective elasticity tensor components. The entire computational methodology is illustrated by two examples––1D and 2D two component periodic composites. The closed form effective Young’s modulus is used in the first example, while the homogenization function is to be computed in the second case. Both illustrations show that different components of the effective elasticity tensor show different sensitivities to particular mechanical properties of the original composite and, further, the illustrations make it possible to determine the most decisive elastic parameters for the homogenization-based computational design studies. Finally, it should be noted that sensitivity analysis can be used for validation of various homogenization methods. In most cases an increase of Young’s moduli of composite components should result in a corresponding increase of the effective material tensor components; the reversed phenomenon can be observed for some specific cases, but usually in an extremely small range only. Therefore, if the sensitivity analysis shows that most of the gradients are negative, the homogenization theory should be essentially corrected. An applied effective modulus method is verified below using the examples of unidirectionally distributed heterogeneities in the periodic two-component bar structure and of a fiber-reinforced periodic composite. As it is demonstrated for plane composite structure, the sensitivity gradients of homogenized elasticity tensor show some instabilities observed for an extremely small value of the perturbation parameter. In the same time, for the Poisson’s ratios values tending to their physical bounds, an uncontrolled increase of all sensitivity gradients is observed. That is why a continuation of this study is necessary in the context of computational error analysis, to extend constitutive models of composite components as well as to evaluate geometrical and material sensitivity gradients for more complex heterogeneous structures, especially in the probabilistic context. Another important topic studied in the paper is the application of the parameter central finite difference analysis to the sensitivity analysis of the uniform plane strain problem of a real composite. It is done under the assumption that the representative volume element (RVE) of a plane cross-section is uniformly extended in two perpendicular directions and the unit shear strain is applied on the RVE. Therefore, the sensitivity functional is proposed as the elastic strain energy stored in the cell that is treated as some type of the representative strain state of a composite under real boundary conditions. To reflect specific conditions of the composite effort more accurately, a particular strain component can be scaled over some multipliers to illustrate pure horizontal and/or vertical extension of the composite specimen. Finally, the sensitivity of this functional is taken as a measure of the influence of various materials parameters on the overall behavior of the composite; according to the previous results, we observe the Poisson’s ratio of the matrix as a dominating material parameter for the fiber-reinforced periodic composites with the RVE specified below. Finally, it should be mentioned that this sensitivity analysis is introduced and performed to validate the homogenization theory itself. In case when the external boundary conditions are known together with the micromorphology of the composite, homogenization theory makes it possible to determine the effective characteristics of this structure and, according to the sensitivity analysis, the sensitivity gradients of both real and homogenized structures are computed. If these gradients have consistent signs and comparable values, then the proposed homogenization algorithm is useful in computational modeling; otherwise another method should be proposed. It can happen that some homogenization theories (or even closed form equations) are valid for some specific boundary value problems only, which can be verified in that way. Another promising field of application of such an analysis is the problem of identification of composite materials and structures.

(1) The sensitivity analysis of homogenized material tensors, proposed and carried out in the paper, makes it possible to compute and analyze the influence of particular material parameters of the composite components on the overall effective properties of a composite. Thanks to such an analysis, a composite designer can generally determine the most decisive material characteristics and then, modifying their values during designing process, can optimize the composite structure for the effective parameters given a priori. The sensitivity equations for homogenization of linear elastic composites can be extended to analogous analysis for effective properties of composites with viscoelastoplastic components, both in deterministic and probabilistic context. The methodology proposed has a relatively general character, however, examination of other engineering composites (beams, plates, various 2 and 3D structures and shells [3] and [18]) can be applied with minor modifications only and should give different results. (2) Numerical analysis performed for two different composites types enabled to detect that fiber-reinforced composites are most sensitive to Poisson’s ratio of the matrix, then to Young’s modulus of the fiber and next––of the matrix. Negative and in the same time the smallest sensitivity is noticed with respect to the Poisson’s coefficient of the fiber. Unidirectional composite structure with the same material properties values corresponding to the glass-epoxy composite showed that fiber Young’s modulus is decisive for this composite, next––Young’s modulus of the matrix. The Poisson’s ratios have smaller influence in this case––the one corresponding to the matrix is positive, while the fiber coefficient results in a negative derivative of the functional G. (3) Detailed computational studies for the fiber-reinforced composite, performed in terms of perturbation parameter ε applied in sensitivity gradients analysis, show that the best numerical stability of all gradients is obtained for ε≈O(−2). Less order taken in numerical analysis causes significantly greater deviations of the final result, while the lack of physical sense of the problem is obtained for ε≈O(−1), where the Poisson’s ratio is taken as a design parameter. Homogenization functions , and visualizations performed in Section 5.2 demonstrated that extremal values in all cases are obtained along the interfaces. Therefore, the most precise mesh should be introduced in the interfaces neighborhood, whereas there is no such necessity along external edges of the RVE. Finally, considering the assumption that the scale factor between the periodicity cell and the entire composite structure tends to 0 and, on the other hand, that this quantity in real composites is small and positive, but differs from 0 (see [22], for instance), the sensitivity of effective characteristics to this parameter is to be calculated next using so-called micro–macro analysis, for instance. To carry out such an analysis, the scale parameter must be inserted in equations describing effective quantities and then the influence of the relation between micro- and macro-structure must be derived explicitly. (4) The sensitivity analysis of 1D homogenization problem carried out in the paper may be applied with some minor modifications in any linear potential field homogenization problem––irrotational and incompressible fluid flow, film lubrication, acoustic vibration as well as for electric conduction, electrostatic field and electromagnetic waves. To use these results for homogenization of other engineering problems, the well-known field analogies [1] may be applied to transform the effective Young’s modulus to related physical fields parameters, i.e. heat conductivity coefficient, seepage permeability, shear modulus, electrostatic permittivity, electric conductivity. Furthermore, the sensitivity studies analogous to those presented above can be carried out in terms of wavelet-based multiscale homogenization of composites. Since the algebraic formulas for the multiresolutional homogenization approach are available now, all the computations can be carried out using any symbolic computations packages and that is why both deterministic and probabilistic [10] and [15] sensitivity analyses may be relatively easily implemented and performed.