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CC2011, Chania, Crete, Greece,6-9 September 2011Thermoelastic behaviour of masonry-like solidswith temperature-dependent Young’s modulus Maria Girardi, Cristina Padovani, Andrea Pagni, Giuseppe Pasquinelli Laboratory of Mechanics of Materials and StructuresInstitute of Information Science and Technologies “A. Faedo” Italian National Research Council Pisa, Italy
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CC2011, Chania, Crete, Greece,6-9 September 2011 Masonry-like solids in the presence of thermal variationsMasonry-like (no-tension) materials are nonlinear elastic materials whose constitutive equation is adoptedto model the mechanical behaviour of solids that do not withstand tensile stresses, such as masonry andstones.There are many engineering problems in which the presence of thermal variations (and then thermaldilatation) must be taken into account Masonry constructions subjected to Refractory linings of converters and ladles employed seasonal thermal variations in the iron and steel industry Δθ ≤ 1600 ºC Δθ = − 40 ºC The thermal variation is so high that the dependence of material constants on temperature cannot be ignored
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CC2011, Chania, Crete, Greece,6-9 September 2011Masonry-like materials under non-isothermal conditionsSym the space of symmetric tensorsSym+ the set of positive semidefinite symmetric tensorsSym- the set of negative semidefinite symmetric tensors the absolute temperature, the reference temperatureE Sym, the symmetric part of the displacement gradient, I the identity tensorT Sym, the Cauchy stress tensor the thermal expansion I, the thermal dilatation due to the the thermal variation the Young’s modulus, the Poisson’s ratio No limitations on = (is small)
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CC2011, Chania, Crete, Greece,6-9 September 2011 For Sym × there exists a unique triple of elements of Sym such that (1)The nonlinear elastic material with stress function = T is called masonry-like material In the absence of thermal variations we get the constitutive equation of masonry-like materials introduced by Heyman and Di Pasquale and Del Piero in the 80s
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CC2011, Chania, Crete, Greece,6-9 September 2011The equations of the thermoelasticity of masonry-like materials are coupledIf we assume thatthen the equations of thermoelasticity are uncoupled and can be integrated separately.The uncoupled equilibrium problem of masonry-like solids with temperature dependentmaterial properties subjected to thermal loads is solved via the finite element code NOSAdeveloped by the Mechanics of Materials and Structures Laboratory for nonlinear structuralanalysis
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CC2011, Chania, Crete, Greece,6-9 September 2011Spherical container made of a masonry-like materialsubjected to thermal loadsLet us consider a spherical container S with inner radius a and outer radius b, made of a masonry-like material in the absence of body forces, subjected to surface tractions and temperatures suchthat the problem has spherical symmetryThe displacement vector and the infinitesimal strain tensor are with The stress tensor and the fracture strain are
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CC2011, Chania, Crete, Greece,6-9 September 2011The solution to the constitutive equation (1) in the case of spherical symmetry is given by If belongs to If belongs to If belongs to If belongs to
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CC2011, Chania, Crete, Greece,6-9 September 2011The equilibrium equation is with the boundary conditions For with we consider the steady temperature distributionwe assume that the thermal expansion is with α positive constantWe want to compare the solutions to the equilibrium problem corresponding to different choicesof the Young’s modulus (with ν=0)
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CC2011, Chania, Crete, Greece,6-9 September 2011If the Young’s modulus does not depend on temperature, then the solution can be calculatedexplicitlyIf the Young’s modulus depends on temperature, then the solution can be calculated numericallyvia the finite element code NOSA
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CC2011, Chania, Crete, Greece,6-9 September 2011 is the transition radiuswhich separates theentirely compressedregion from the crackedregion Cracked region =0 <0 =0 >0 compressed region <0 <0 =0 =0
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CC2011, Chania, Crete, Greece,6-9 September 2011
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CC2011, Chania, Crete, Greece,6-9 September 2011
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CC2011, Chania, Crete, Greece,6-9 September 2011 Masonry-like material
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CC2011, Chania, Crete, Greece,6-9 September 2011 Masonry-like material
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CC2011, Chania, Crete, Greece,6-9 September 2011 Masonry-like material
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CC2011, Chania, Crete, Greece,6-9 September 2011 Masonry-like material
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CC2011, Chania, Crete, Greece,6-9 September 2011
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CC2011, Chania, Crete, Greece,6-9 September 2011
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CC2011, Chania, Crete, Greece,6-9 September 2011
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CC2011, Chania, Crete, Greece,6-9 September 2011ConclusionsIn order to model numerically the behaviour of masonry-like solids in the presence of thermal variationsit is fundamental1. To have realistic constitutive equations for the material2. To know the dependence of the material constants on temperature (experimental data)Further investigation is necessary to assess how temperature-dependent material properties influence thestress field and the crack distribution in the case of thermomechanical coupling
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