Some new thermo-elastic solutions for cylindrical and spherical composites
Creators: name: Carannnante, Federico
Publisher: name: Università degli Studi di Napoli Federico II
Year: 2013

description: In modern engineering applications, multilayered structures are extensively used due to the added advantage of combining physical, mechanical, and thermal properties of different materials. Many of these applications require a detailed knowledge of transient temperature and heat-flux distribution within the component layers. Both analytical and numerical techniques may be used to solve such problems. Nonetheless, numerical solutions are preferred and prevalent in practice, due to either unavailability or higher mathematical complexity of the corresponding exact solutions.
Rather limited use of analytical solutions should not diminish their merit over numerical ones; since exact solutions, if available, provide an insight into the governing physics of the problem, which is typically missing in any numerical solution. Moreover, analyzing closed-form solutions to obtain optimal design options for any particular application of interest is relatively simpler. In addition, exact solutions find their applications in validating and comparing various numerical algorithms to help improve computational efficiency of computer codes that currently rely on numerical techniques. Although multilayer heat conduction problems have been studied in great detail and various solution methods including orthogonal and quasi-orthogonal expansion technique, Laplace transform method, Green’s function approach, finite integral transform technique are readily available; there is a continued need to develop and explore novel methods to solve problems for which exact solutions still do not exist. One such problem is to determine exact unsteady temperature distribution in polar coordinates with multiple layers in the radial direction.
Numerous applications involving multilayer cylindrical geometry require evaluation of temperature distribution in complete disk-type. One typical example is a nuclear fuel rod, which consists of concentric layers of different materials and often subjected to asymmetric boundary conditions.
Moreover, several other applications including multilayer insulation materials, double heat-flux conductimeter, typical laser absorption calorimetry experiments, cryogenic systems, and other cylindrical building structures would benefit from such analytical solutions.
Then, object of the present thesis is to derive new thermo-elastic solutions for composite materials constituted by multilayered spheres and cylinders under time-dependent boundary conditions. These solutions are utilized for several engineering applications and we report some applications in last analyze chapters of present thesis. In follows, we will described the contents of thesis. 
In first chapters are reported the thermo-mechanical foundations and a summary of the formulation of thermo-elastic problems for isotropic material.  
In chapter X it is developed an analytical approach to find exact elastic solutions for multilayered cylinder composed of isotropic constituents and determining the analytical response in terms of displacements and stresses for all the De Saint Venant (DSV) load conditions, that is axial force, torque, pure bending and combined bending moment and shear actions. Successively, on the basis of the found analytical solutions, a homogenization procedure is adopted in order to obtain the overall constitutive elastic laws for multilayered cylinder, in this way deriving the exact one-dimensional model characterized by the axial stiffness, flexural rigidity, shear deformability and torsional stiffness relating beam’s generalized stresses and strains. By playing with the Poisson ratios of adjacent phases, some counterintuitive and engineering relevant results are shown with reference to unexpected increasing of overall stiffness of multilayered cylinder.
In chapter XI it is presented an analytical elastic solution for multilayered cylinder constituted by transversally-isotropic n-phases, under radial pressure, axial force and torque. Then, by utilizing the homogenization theory, it is obtained the overall elastic stiffness of the equivalent homogeneous transversally-isotropic solid, establishing the constitutive elastic laws relating stresses and strains.
In chapter XII it is developed an analytical approach to find exact elastic solutions for multilayered cylinder subjected to axial force, constituted by n orthotropic cylindrical hollow phases and a central core, each of them modelled as homogeneous and cylindrically anisotropic material. 
In chapter XIII it is reported an analytical solution for multilayered cylinder composed by hollow cylindrical monoclinic phases under axial force and torsion. In this chapter, we consider the chiral structure for each cylindrical layer. In particular the composite material is constituted by two hollow cylindrical monoclinic phases. The cylindrical monoclinic elastic property of multilayered cylinder is obtained by the particular chiral structure. In fact, we consider the two hollow phases constructed by  right-handed and left-handed spiral helices whose long axes are all parallel. These helical spirals may be either touching or separated by a matrix material and are composed by elastic orthotropic material. 
In chapters XIV, XV, XVI are reported some thermo-elastic solution, for hollow cylinders, hollow spheres and plates, respectively. 
In chapter XVII we consider a steady-state thermo-elastic problem of multilayered cylinder with finite length. The thermal and mechanical loads applied on the cylinder are axisymmetric in the hoop direction and are constant in the axial direction. In order to obtain analytical solutions for temperature, displacements, and stresses for the two-dimensional thermo-elastic problem, the cylinder is assumed to be composed of n fictitious layers in the radial direction. The material properties of each layer are assumed as homogeneous.
In chapter XVIII are determined the displacements, strains, and stresses from the general analytical solution of multilayered sphere composed by an arbitrary number of layers constituted by materials with generic modulus of elasticity, thermal expansion coefficient and thermal conductivity. Material properties are assumed to be temperature-independent and homogeneous in each layer. The multilayered sphere is considered as a classical composite material whose properties abruptly vary from one hollow sphere to the other.
In chapter XIX are presented the most important standard fire curves: ISO 834, External fire curve, hydrocarbon fire curve, ASM119 and parametric fire curves (European Parametric fire curves, Swedish Fire Curves, BFD curves, CE 534 curve). Moreover in this chapter are reported the mechanical and thermal properties of steel and concrete at elevate temperature.
In chapters XX and XXI, the one-dimensional quasi-static uncoupled thermo-elastic problem of a multilayered sphere and multilayered cylinder, with time-dependent boundary conditions are considered, respectively. The body forces and heat generation vanish. In both cases, the analytical solution is obtained by applying the method of separation of variables.
In chapter XXII it is studied a spherical tank methane gas-filled exposed to fire characterized by hydrocarbon fire curve. The interaction between spherical tank and internal gas is studied. By applying a suitable simplified hypothesis on the mechanics of problem, we determine the analytical thermo-elastic solution for spherical tank. By applying the solution obtained, the increasing graded temperature of gas methane in spherical tank is determined. Finally, a numerical example is reported for a spherical tank exposed to hydrocarbon fire, showing the collapse temperature.
In chapter XXIII, an industrial insulated pipeline is modelled as multilayered cylinder, subjected to mechanical and thermal loads. By using a multi-layered approach based on the theory of laminated composites, the solutions for temperature, heat flux, displacements, and thermal/mechanical stresses are presented. By applying the analytical thermo-elastic solution reported in Chapter XVII, a parametric analysis is conducted in order to analyze the mechanical behaviour of an industrial insulated pipeline composed by three phases: steel, insulate coating, and outer layer made of polyethylene to protect the insulation. In this model, parametric analyses are conducted by varying the Young’s modulus, Poisson’s ratio, thermal conductivity and linear thermal expansion coefficient of insulate coating. The analysis shows the maximum Hencky von Mises’s equivalent stress in steel phase and in insulate coating.  Finally, it is presented a numerical example by considering three types of materials for insulate coating: (1) Expanded Polyurethane; (2) Laminate glass; (3) Syntatic foam. 
In chapter XXIV it is analyzed a cylindrical concrete specimen under axial force within Fibre Polymeric Reinforcing sheets. The elastic solutions found in Chapter XII are here extended to the post-elastic range. The evolution of the stress field when the core phase is characterized by an Intrinsic Curve or Schleicher-like elastic-plastic response with associate flow rule and the cylindrically orthotropic hollow phase obeys to is shown the elastic-brittle Tsai-Hill anisotropic yield criterion. The choice of these post-elastic behaviours is suggested by experimental evidences reported in literature for these materials, as well as the cylindrical orthotropy of the hollow phase intrinsically yields to consider several perfectly bonded FRP layers as an equivalent one, interpreting their overall mechanical response by invoking the theory of homogenization and the mechanics of composites. 
At the end, a numerical example application to cylindrical concrete specimens reinforced with Carbon FRP is presented, by furnishing a predictive formula – derived from the previously obtained analytical solutions - for estimating the overall collapse mechanism, the concrete ultimate compressive strength and the confining pressure effect. The results are finally compared with several experimental literature data, highlighting the very good agreement between the theoretical predictions and the laboratory measurements.
In chapter XXV it is reported an analytical thermo-elastic solution in closed form for bi-layer hollow cylinder subjected to time-dependent boundary conditions. It is assumed that each hollow cylinder is composed by a homogeneous and thermo-isotropic material, characterized by different mechanical and thermal parameters, i.e. modulus of elasticity, thermal expansion coefficient and thermal conductivity. Moreover, these material properties in each hollow cylinder are assumed to be temperature-independent. In other words, the bi-layer hollow cylinder is considered as a classical composite material whose properties abruptly vary from one hollow cylinder to the other. In particular, it is obtained a new analytical solution for a bi-layer hollow cylinder, constituted by two phases: Ceramic ( ) and Metal ( ) subjected to heat flux on inner surface.