Multiscale modelling of thermal conductivity in composite materials for cryogenic structures A. Alzina a,b, * , E. Toussaint b , A. Be ´akou b , B. Skoczen a a European Organisation for Nuclear Research, Geneva, Switzerland b Laboratoire de Me ´canique et Inge ´nieries, Campus de Clermont-Ferrand/Les Ce ´zeaux, 63170 Aubiere, France Available online 14 June 2005 Abstract A multiscale analysis is performed to estimate overall thermal conductivity in a column-type support post used in particle accel- erators to sustain cryomagnets. A non-linear homogenisation of thermal conductivity in a unidirectional composite material includ- ing phonon mechanism is firstly proposed by using the two-scale asymptotic expansion method. Next, computed conductivity curves are used for the modelling of heat transfer in a global representative volume element of the braided fabric composite. Thermal tests are conducted to measure conductivity of the material. A good agreement is obtained between estimated and measured values. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Polymer-matrix composites; Thermal properties; Numerical analysis; Thermal analysis; Cryogenic temperatures 1. Introduction The large Hadron collider (LHC) under construction at the European Organisation for Nuclear Research makes use of 1700 cryomagnets for orbital bending or focusing/defocusing of the circulating high-energy pro- ton beam. Cylinder-type support posts are used to sus- tain these cryomagnets and achieve thermal insulation (Fig. 1a). Their thermal [1] and mechanical requirements dictate the materials selection. An example of perfor- mance index used is the ratio I ¼ r qk , the specific strength per average thermal conductivity which must be high. Typical values of ratio I are given in Ref. [2] for metals and polymer matrix composites in the 4.2–77 K temper- ature range. It appears that for composite materials, this ratio is many times higher than for metallic material. Consequently, organic composite materials are attrac- tive alternatives to metals for the design of the support posts as they allow substantial savings in refrigeration costs. Composite support posts may be either filament wound, pultruded in a continuous process or high-pres- sure laminated with a fabric reinforcement. It is obvious that the spacial arrangement of the reinforcement and their size strongly influence the engineering properties of the support posts. The material used in the LHC pro- ject is a triaxially braided composite with a longitudinal yarn and two braid yarns at ±45° (Fig. 1b). Each yarn contains thousands of 6–9 lm diameter fibres. Thus the computational size of the full-scale problem exceeds the capacity of any available computational system. Introducing homogenisation method as a multiscale analysis substantially decreases the computational size of the problem by decoupling the scales into a micro and macroscale. Hierarchical or multiscale analysis attempts to incor- porate fine-scale features in a global analysis using var- ious techniques: multigrid methods [3], Voronoi-cell finite elements [4] or estimation of modelling error [5]. These approaches are usefull for very sensitive applica- tions where local behaviour is needed and could not 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.04.002 * Corresponding author. Address: European Organisation for Nuclear Research, Geneva, Switzerland. Fax: +33 473 288 100. E-mail address: arnaud.alzina@cern.ch (A. Alzina). Composite Structures 74 (2006) 175–185 www.elsevier.com/locate/compstruct