First A. Author, Second B. Author and Third C. Coauthor.

Four-scale bridging for the non linear analysis

of composites including continuum to discrete linkage

Bernhard A. Schrefler *, Daniela P. Boso* and Marek Lefik†

* Department of Structural and Transportation Engineering

University of Padua

Via Marzolo 9, 35131 Padua, Italy

E-mail: ,

† Chair of Geotechnical Engineering and Engineering Structures

Technical University of Łódz

Al.Politechniki 6, 93-590 Lodz, Poland

E-mail:

Key words: Periodic Composites, Multiscale Analysis, Asymptotic Homogenization, Finite Element Method.

Summary. In this paper the multiscale analysis of hierarchical composite with periodic microstructures is presented. The method is illustrated via an applicative example: the multiscale analysis of a SC cable designed for the International Thermonuclear Experimental Reactor (ITER) is discussed. The classical theory of asymptotic homogenization together with the Finite Element Method is used and extended to obtain the non-linear, temperature dependent material characteristics of the components. Four scales are identified for the example, and at an intermediate scale the mechanics is no more continuous, but becomes discrete. The continuum-to-discrete linkage is thus realized, permitting the analysis at global level via a continuum model.

1 INTRODUCTION

Many advanced applications require the use of engineered material microstructures which necessarily involve the use of multiphase material concepts. Since starting materials have different properties, it seems rational to combine them to obtain a tailored behaviour as a final result. Usually composite materials are made of a matrix which could be metal, polymeric or ceramic, and one or more reinforcements, which could be particles or fibres. Each phase plays its own role on the global response. To study material and structural behaviour at a macroscopic level single scale models are mainly used. However, many of these natural and man-made materials exhibit an internal structure at more than one length scale. These internal structures may be of a translational nature, where the structure is more or less invariant with respect to a translation corresponding to the smallest length scale. Materials with internal structure may show also multiscale features, i.e. they may be invariant with respect to scaling. Such materials can be considered to be fractal-like, but they are not true fractals since the exponent n remains finite and the volume fraction does not go to zero even for large n.

Beyond material behaviour the overall structural behaviour is also of interest. It is at this level that external loads are usually specified and the structural response has to be determined. At the structural level a direct simulation of multiscale systems is usually rather complex and time consuming because the discretization has to be reduced to the lowest scale at which information is needed. This would allow one to obtain at the same time the macroscopic behaviour of the entire structure as well as all the information at the lower scales. But this is not necessary and in a case where we start from an atomic or a nano-scale level is not yet possible. Multiscale modelling which links the different single-scale models in a hierarchical way is an answer to the problem. In the case of material multiscale modelling it is usually of interest to proceed from the lower scales upward in order to obtain homogenized material properties. Alternatively, in the case of structural modelling it is important to be able to step down through the scales until the desired scale of the real, not homogenized, material is reached. This technique is often known as unsmearing or localization. Usually in a global analysis both aspects need to be pursued.

Figure 1. Hierarchical structure of ITER cable. It is assembled according to a multilevel twisting process: thousands of SC filaments (upper row, on the left) are embedded in a resistive matrix to obtain the strand (upper row, on the right). Then three strands are twisted into a triplet, then four triplets are again twisted to obtain a bundle, four of these bundles are twisted again to obtain a higher order bundle (lower row, on the right), then four of them are twisted together to obtain the last but one cabling stage: the petal. Six petals (lower row, on the centre) are then twisted around a spiral tube and inserted into a jacket to obtain the final cable.

As an application of multiscale modelling we present here an analysis of a Nb3Sn based superconducting (SC) cable designed for the International Thermonuclear Experimental Reactor (ITER). This is an excellent example of a hierarchical structure, where lower levels take part in the global behaviour. In fact according to the current design, the SC alloy is formed into fine filaments, which are embedded in a low-resistivity matrix of normal metal to make the elementary strand. After that more than one thousand strands are twisted together according to a multi-level twisting scheme to form the final cable (Fig. 1).

2 Statement of the problem and assumptions

Since Nb3Sn based strands are strain sensitive, it is extremely important to know the strain field under operating conditions. Because of the scale separation between structural levels a spatial discretization - e.g. that of a finite element mesh - fine enough for the micro level would result in a huge number of elements and unknowns at the macro level. This would be numerically difficult to manage and, first of all, not necessary. As mentioned in the introduction an alternative approach is here proposed: the macroscopic behaviour is studied by means of a homogenized constitutive relation.

The SC cable is characterized by a regular or nearly regular structure, so that we can assume a periodic structure for this kind of composite. In this case it is possible to obtain the equivalent homogenised material exploiting the theory of homogenisation. From a mathematical point of view, the theory of homogenisation is a limit theory which uses the asymptotic expansion and the assumption of periodicity to substitute the differential equations with rapidly oscillating coefficients, with differential equations whose coefficient are constant or slowly varying in such a way that the solutions are close to the initial equations1,2,3.

In the SC cable we can distinguish four scales:

·  the filament (about 4 mm diameter);

·  the filament group (about 50 mm diameter);

·  the strand (about 0.8 mm diameter);

·  the cable (macro scale, about 5 cm diameter).

It is worth to point out that in the passage from the third to the fourth level there is also a transit from continuum to a discrete model (strand → triplet).

We start the procedure at the first level with estimation of the effective mechanical and thermal coefficients of the composite4. This is done by an analysis of the micro-cell of periodicity, according to the classical theory of homogenisation and via the Finite Element method. These two tools are here extended for the piecewise linear analysis of the SC fibrous composite with non-linear, temperature dependent components. We account also for local material yielding at the stage of microanalysis. To recover the strain inside each single component a suitable unsmearing technique is then applied. The procedure for the second level is the same, and it gives the effective coefficients to be used for the strand.

At the last but one scale a fibrous beam model is used5,6, which inherits the properties of the composite calculated at the previous levels. For the subsequent scale bridging we choose as unit cell a triplet of strands, which is the first cabling stage (Fig. 1). At this level the problem is different: the mechanics is no more continuous, but becomes discrete. As a consequence in the definition of the effective properties at this level we take into consideration the contact phenomena occurring between the strands7.

Going to the macro level, the SC cable or a sub-cable stage is analysed using the thermo-mechanical properties calculated at meso level. In this way we can analyse a bundle of strands via a continuum model thus realizing the continuum-to-discrete linkage.

Acknowledgements

Support for this work was partially provided by PRIN 2004094015_002 “Thermo-hydraulic-mechanical and Electro-mechanical Modelling of ITER Superconducting Magnets” and by “KMM-NoE - Knowledge-based multicomponent materials for durable and safe performance - Network of Excellence”. This support is gratefully acknowledged.

REFERENCES

[1]  A. Bensoussan, J. L. Lions, G. Papanicolau. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1976.

[2]  E. Sanchez-Palencia. Non-Homogeneous Media and Vibration Theory. Springer V. Berlin, 1980.

[3]  B. Hassani, E. Hinton. A review of homogenization and topology optimization I-homogenization theory for media with periodic structure, Computer and Strucutres, 69, 707-717, 1998.

[4]  D.P. Boso, M. Lefik, B.A. Schrefler “A multilevel homogenised model for superconducting strands thermomechanics”, Cryogenics, 45/4, 259-271, 2005.

[5]  M. Lefik, B.A., Schrefler, 3D finite element analysis of composite beams with parallel fibres based on the homogenization theory, Computational Mechanics, 14, 1, 2-15, 1994.

[6]  D.P. Boso, M. Lefik, B.A. Schrefler. Multiscale Analysis of the Influence of the Triplet Helicoidal Geometry on the Strain State of a Nb3Sn Based Strand for ITER Coils, in print on Cryogenics.

[7]  H.W. Zhang, D.P. Boso, B.A. Schrefler. Homogeneous analysis of periodic assemblies of elastoplastic disks in contact, Int. J. of Multiscale Computational Engineering, 1(4), 349-370, 2003.

3