A numerical method for solving equilibrium problems of no-tension solids subjected to thermal loads Cristina Padovani * , Giuseppe Pasquinelli, Nicola Zani Consiglio Nazionale delle Richerche, Istituto CNUCE-CNR, Via Santa Maria 36, 56126 Pisa, Italy Received 8 September 1998; received in revised form 27 January 1999 Abstract This paper starts out by recalling a constitutive equation of no-tension materials that accounts for thermal dilatation and the temperature dependence of the material parameters. Subsequently, a numerical method is presented for solving, via the ®nite element method, equilibrium problems of no-tension solids subjected to thermal loads. Finally, three examples are solved and discussed: a spherical container subjected to two uniform radial pressures and a steady temperature distribution, a masonry arch subjected to a uniform temperature distribution and a converter used in the steel and iron industry. Ó 2000 Elsevier Science S.A. All rights reserved. Keywords: No-tension materials; Thermal loads; Finite element method 1. Introduction In many applications it is necessary to model the behaviour of solids not withstanding tension in the presence of thermal dilatation. For example, molten metal production processes, in particular integrated steel manufacturing, require refractory linings able to withstand the thermo-mechanical actions produced by high-temperature ¯uids [1]. Analysis of these coverings is usually carried out by considering the re- fractory materials to be linear elastic, exhibiting the same behaviour in the presence of tension and com- pression, though they are actually non-resistant to traction. Results obtained by applying such a constitutive model are generally characterised by considerable tensile stresses and are thus quite unrealistic. However, there are many other engineering problems concerning no-tension solids in which thermal dilatation must be accounted for: consider, for example geological problems connected with the presence of a volcanic caldera, such as that of Pozzuoli [2], or the in¯uence of thermal variations on stress ®elds in masonry bridges [3]. In many such cases the thermal variation is so high that the dependence of the material parameters on temperature cannot be ignored. In [4] the authors present a constitutive equation for isotropic no-tension materials in the presence of thermal expansion which accounts for the temperature-dependence of the material's parameters. In par- ticular, explicit expressions for stress and inelastic strain are given as functions of the strain minus the thermal dilatation; from these free energy, internal energy and entropy are then obtained, and both coupled and uncoupled equations of the thermo-mechanical equilibrium of a no-tension solid have been developed. In this paper we recall the constitutive equation presented in [4] and, by limiting ourselves to thermo- mechanical uncoupling, we propose a numerical method for solution of the equilibrium problem of solids not supporting tension that are subjected to thermal loads. www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 190 (2000) 55±73 * Corresponding author. 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 3 4 6 - 1