Adriano Montanaro

On singular surfaces in isotropic linear thermoelasticity with initial stress

It will be shown that the theory of a fundamental paper of Chadwick and Powdrill on singular surfaces, propagating in a linear thermoelastic body which is stress-free, homogeneous, and isotropic, also holds when the medium is subjected to hydrostatic initial stress provided the two characteristic speeds are suitably changed. The result is obtained by using Biot’s linearization of the constitutive law for the stress.

Wave propagation along axes of symmetry in linearly elastic media with initial stress

The proofs of the Fedorov–Stippes and Fedorovs theorems, which hold for linearly elastic homogeneous bodies in natural configurations, remain valid for any linearly elastic medium with initial stress provided that Hookes tensor be replaced by a suitable elasticity tensor.

Constitutive equations and wave propagation in Green–Naghdi type II and III thermoelectroelasticity

ABSTRACT In this article we extend the theory of thermoelasticity devised by Green and Naghdi to the framework of finite thermoelectroelasticity. Both isotropic and transversely isotropic bodies are considered and thermodynamic restrictions on their constitutive relations are obtained by virtue of the reduced energy equality. In the second part, a linearized theory for transversely isotropic thermopiezoelectricity is derived from thermodynamic restrictions by constructing the free energy as a quadratic function of the 11 second-order invariants of the basic fields. The resulting theory provides a natural extension of the (linear) Green and Naghdi theory for types II and III rigid heat conductors. As a particular case, we derive the linear system which rules the processes depending on the symmetry axis coordinate only.

On heat flux in simple media

For any simple body we prove that there is an infinite number of mathematical functions which can be added to the response functional for the heat flux vector without affecting the balance laws, the entropy inequality, and the boundary conditions on the normal heat flux. The presence of this indetermination cannot be detected by usual physical experiments, namely by experiments in which cuts of the body are not taken into account. The maximal class of this indetermination is fully characterized in the thermoelastic case.

On discontinuity waves in linear piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation.

On a large class of symmetric systems of linear PDEs, for tensor functions, useful in mathematical physics

We study a class of symmetric systems of linear partial differential equations which involve tensor functions relating tensor spaces on a three dimensional vector space, on the real field, equipped with an inner product. These systems arise by coupling certain simpler symmetric systems studied in a previous paper. In order to investigate some questions, related to constitutive equations for bodies of the differential type, certain classes of physically privileged solutions are characterized for some of the aforementioned systems.

A dissipation inequality involving only the dynamic part of entropy

Abstract For a general body B of the differential type and arbitrary complexity, we set up a thermodynamic theory T in which only the dynamic part of entropy is assumed as primitive. Indeed, in T the existence of the equilibrium entropy is not assumed and, furthermore, the dissipative inequality involves only the dynamic part of entropy. By a certain Gibbs relation proved here, a magnitude ν∗, to be called rate of change of the equilibrium entropic, is denned by means of equilibrium stress power and equilibrium internal energy, without having at our disposal a response function for the equilibrium entropie. This definition agrees with the equality which yields the rate of change of the equilibrium entropy in the corresponding classical theory T based on the Clausius-Duhem inequality and in which entropy is a primitive. The well-posedness of the definition given for ν*, from the physical point of view, is assured both by the uniqueness theorem for the response function of the stress and by the uniqueness theorem for the response function of the equilibrium internal energy, proved here for any complexity of the material. In T the Clausius-Duhem inequality is stated in terms of ν∗ and holds as a theorem. We show that the class of constitutive functions which represent a material in T is strictly larger than the analogous class in T . Indeed, in the former class the equilibrium entropic may have no response function, whereas in the latter class obviously the equilibrium entropy always has a response function. Furthermore, each material in T is a material in T too. Hence, theory T is more general than theory T . The existence of a response function for the equilibrium entropic is equivalent to a certain integrability condition, regarding a system of PDEs involving the equilibrium response functions of the stress and of the internal energy. By postulating this condition, we obtain a theory for which there exists a response function for the equilibrium entropic and such that any theorem of the classical theory T holds. In this case entropic can be called entropy, as in T .

Entropy-free theories for differential materials

Abstract A thermodynamic theory was set up for general bodies of the differential type and complexity one without postulating entropy as a primitive. Indeed, a theory was proposed for such bodies in which the entropy is not postulated to exist from the outset, and in which the Clausius-Duhem inequality is replaced by a dissipative inequality involving stress, internal energy and heat flux. Entropy rate-of-change , v , is denned by means of equilibrium stress and internal energy without having at disposal a response function for the entropy, and shows that a certain condition of integrability regarding the response functions of the stress and internal energy is equivalent to the existence of a response function for the entropy. Hence, by postulating this condition to hold within the theory proposed here, we can prove the Clausius-Duhem inequality to hold, and thus also all the theorems of the corresponding classical theory. Any argument above holds in the particular case of an elastic body.

Propagation of Wavefronts in Thermoelastic Media

This paper proposes a fully-coupled thermomechanical analysis of multilayered plates and shells. In the proposed refined plate/shell models, the temperature is considered as a primary variable of the problem as the displacement and it is directly obtained from the governing equations. Such models are very promising for multilayered structures because they permit both equivalent single layer and layer wise approaches and they have the order of expansion in the thickness direction as a free parameter (from linear to fourth order). Three different problems can be analyzed: evaluation of temperature field effects in the free vibration analysis of multilayered plates and shells; evaluation of temperature field effects in the stress analysis of multilayered structures subjected to mechanical loads; thermal stress analysis of multilayered structures with imposed sovra-temperature.From the Contents: - Aero-Thermo-Elasticity, Aero-Magneto-Elasticity, Functionally Graded Plates, Shells and Beams.- Analytical (Computational) Thermomechanics.-Ceramics and Linear Fracture Mechanics.-Composites.-Contact Problems.-Coupled and Generalized Thermoelasticity.-Elastostatics and Thermoelastostatics.-Electro-Elastic Thermoelasticity and Smart Structures.-Electronics, Optoelectronics, Photonics, and MEMS (MOEMS) Packaging Engineering.-Fracture.-Heat Conduction-Direct, Inverse and Optimization.-Heat Treatment, Welding, and Shape Memory.-Linear and Nonlinear Viscoelasticity and Viscoplasticity.-Mathematical Preliminaries and Methods.-Methods of Complex Variables.-Nanotechnology, Special Methods in Thermal Stresses.-Plates.-Qualitive Properties of Thermoelastic Solutions.-Shells.-Simulation and Modeling-Thermal Stresses.-Stability.-Thermal Stress Resistance, Experimental Methods.-Thermal Stresses-Basic Problems.-Thermal Stresses Induced by Laser Heating.-Thermo-Inelasticity and Damage.-Thermodynamics of Thermodeformable Solids.-Thermoelastodynamics.-Transient Thermoelastic Waves and Dynamic Problems.-

On discontinuity waves and vibrations in thermo-piezoelectric bodies

With regard to a body composed of a linear thermo-piezoelectric medium, referred to a natural configuration, we consider processes for it constituted by small displacements, thermal deviations, and small electric fields superposed to the natural state. We show that any discontinuity surface of order r greater than 1 for the above processes is characteristic for the linear thermo-piezoelectric partial differential equations. We show that discontinuity surfaces of order 0 generally are not characteristic; hence, the conditions are written, which characterize the discontinuity surfaces of order 0 that are characteristic. We find the ordinary differential equations of propagation for plane progressive waves and standing waves. Then we characterize the ones whose wavefronts are characteristic.