Keywords: sensitivity analysis, composite materials, homogenisation technique, finite element method, Monte-Carlo simulation, probabilistic mechanics.

Homogenisation method serves for determination of the effective parameters for heterogeneous and composite materials [1,2,3,4]. One of the intermediate effects of the homogenisation approach is elimination of some structural parameters from original composite model. This is a reason to verify the sensitivity gradients of effective composite and to compare them in some engineering structural problems with the sensitivity of real composites. Such an analysis can validate the specific homogenisation method - it can be approved when the sensitivity coefficients for real and homogenised composites have at least the same signs and, preferably, comparable values. The entire approach to the sensitivity analysis in homogenisation slightly differs from classical approach known from the structural analysis applications [1]. The relevant gradients do not concern each degree of freedom separately but, arbitrarily, specific components of the effective elasticity tensor. Analogous facts take place in case of random composites - instead of design parameters, their probabilistic moments and coefficients give the input to numerical studies. It significantly extends both computational implementation and the analysis - each output probabilistic moment is differentiated with respect to any probabilistic moment of the composite input, so that final results are significantly more complicated than in deterministic case.

The effective modulus homogenisation method is a core of the presented computations. Sensitivity gradients cannot be obtained analytically, because the homogenisation function components are determined numerically by only as some cell problems solutions. Generally, two separate ways can be followed - pure computational finite difference based studies and semi-analytical method, where spatial averages of the constitutive tensor components are differentiated symbolically and the remaining part is treated using straightforward differentiation technique. Taking into account the consistency with the Monte-Carlo simulation application [2] and computational time savings, full numerical differentiation has been implemented in the special purpose FEM code.

The main goal in numerical analysis was to determine numerically the increment of the effective parameters for four-component composite resulting from the additional increments of some input material parameters. The sensitivity coefficients for the effective elasticity tensor are computed in deterministic case w.r.t. design parameters vector h as [1]

This definition can be extended on random composites, the sensitivity coefficients of m-th order probabilistic moment of this tensor with respect to n-th order central probabilistic moment of input random design variable vector h can be determined.

Engineering application of the computational methodology is illustrated with the example of periodic superconducting coil cable (Figure 63.1a), where material parameters are defined using deterministic quantities and, separately, expected values and standard deviations [2]. Homogenisation is carried out using the special-purpose program MCCEFF and 4-node rectangular isoparametric finite elements (see Figure 63.1b). Elastic properties of the superconductor and the jacket were determined as decisive for the overall plane strain problem for this composite. The results obtained can be used further in optimisation of the superconducting composite structures [4].

1

M. Kaminski, "Sensitivity analysis of homogenized characteristics for some elastic composites". Computer Methods in Applied Mechanics & Engineering, 192(16-18), 1973-2005, 2003. doi:10.1016/S0045-7825(03)00214-7

2

M. Kaminski, B.A. Schrefler, "Probabilistic effective characteristics of cables for superconducting coils". Computer Methods in Applied Mechanics & Engineering, 188(1-3), 1-16, 2000. doi:10.1016/S0045-7825(99)00424-7

3

G.W. Milton, "The Theory of Composites". Cambridge University Press, Cambridge, 2002.

4

P. Pedersen, ed., "Optimal Design with Advanced Materials". Elsevier, 1993.

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