Generalized thermomechanics with dual internal variables

The formal structure of generalized continuum theories is recovered by means of the extension of canonical thermomechanics with dual weakly non-local internal variables. The canonical thermomechanics provides the best framework for such generalization. The Cosserat, micromorphic, and second gradient elasticity theory are considered as examples of the obtained formalization.     

Geometrically nonlinear continuum thermomechanics with surface energies coupled to diffusion

Surfaces can have a significant influence on the overall response of a continuum body but are often neglected or accounted for in an ad hoc manner. This work is concerned with a nonlinear continuum thermomechanics formulation which accounts for surface structures and includes the effects of diffusion and viscoelasticity. The formulation is presented within a thermodynamically consistent framework and elucidates the nature of the coupling between the various fields, and the surface and the bulk. Conservation principles are used to determine the form of the constitutive relations and the evolution equations. Restrictions on the jump in the temperature and the chemical potential between the surface and the bulk are not a priori assumptions, rather they arise from the reduced dissipation inequality on the surface and are shown to be satisfiable without imposing the standard assumptions of thermal and chemical slavery. The nature of the constitutive relations is made clear via an example wherein the form of the Helmholtz energy is explicitly given.     

On canonical equations of continuum thermomechanics

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct is made of necessity for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including pure elasticity, inhomogeneous thermoelasticity of conductors, and the case of dissipative solid-like materials described by means of a diffusive internal variable (such as in damage or weakly non-local plasticity). A redefinition of the entropy flux is necessarily accompanied by a redefinition of the Eshelby stress tensor.     

A virtual power format for thermomechanics

It is shown that a virtual power format slightly more general than usual may be employed to deduce all balance and imbalance laws of thermomechanics. An essential role is played by the notion of thermal displacement; the basic balance laws turn out to be those for momentum and entropy. In consequence of these balances and of two axioms of thermodynamical nature—namely, conservation of internal action in cyclic processes and dissipative nature of ordinary processes—balance of energy and inbalance of entropy are arrived at. Dedicated to Tommaso Ruggeri, on the occasion of his 60th birthday.     

Fundamentals of thermomechanics of porous saturated media

Institute of Mechanics, Academy, of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 4, pp. 3–13, April, 1988.     

Two-temperature continuum thermomechanics of deforming amorphous solids

There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.     

Remarks on the Eshelbian thermomechanics of materials

The notion of Legendre–Fenchel transformation is used jointly with that of continuum mechanics on the material manifold (so-called Eshelbian mechanics) in order to specify more easily the relevant thermodynamical regime. This has direct consequences in the formulation of the appropriate driving force acting on various singular surfaces. Thanks to the notion of material “thermal” force, the formalism also provides directly a proof of theorems such as those of Vazsonyi–Crocco, Helmholtz, and Bernoulli.     

Mathematical models of thermomechanics of a relaxing solid

Mathematical models of thermomechanic processes based on the laws of rational thermodynamics of irreversible processes are considered. Specific characteristics of the continuum nonstationary behavior are shown in the framework of variousmodels of a medium with internal state parameters.     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

Condensation with interfacial temperature drop

Using the boundary layer equations of momentum and energy, an asymptotic series has been developed to determine the velocity and temperature fields in a stationary metal vapor condensing over a vertical surface, with the film in laminar flow. Interfacial temperature drop has been taken into account, by using an approximate form of the developed byschrage to predict the condensate metal surface temperature. With two terms taken from the asymptotic series development, it is shown that the resulting values of heat transfer coefficients and average Nusselt numbers fall in the same range as the experimentally observed values for mercury vapor. These calculations have been carried out assuming a condensation coefficient of unity. The Nusselt number is seen to depend parametrically on both the saturation pressure and the length of the condensing surface. The calculated mass condensation rate is lower than that of the theory neglecting interfacial temperature drop. The results are all applicable for sufficiently long plates when the pressure is not too low. For very short plates and pressures below about 25 mm of mercury, more terms in the asymptotic expansion must be considered.

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Zusammenfassung Unter Benutzung der Grenzschichtgleichungen für Impuls und Energie wird eine asymptotische Eeihenentwicklung vorgenommen, um die Geschwindigkeits und Temperaturfelder in einem stationären Metalldampf, welcher an einer senkrechten Oberfläche kondensiert, zu bestimmen; der Film befindet sich dabei in laminarer Strömung. Das Temperaturgefälle an der Grenzfläche wurde berücksichtigt durch Benutzung einer Näherungsform der vonschrage entwickelten Gleichung, aus der sich die Oberflächentemperatur des kondensierenden Metalls ableiten läßt. Mit den beiden ersten Termen der asymptotischen Reihenentwicklung wird gezeigt, daß die resultierenden Werte der Wärmeübertragungskoeffizienten und der durchschnittlichen Nusselt-Zahl in denselben Bereich fallen wie die experimentell beobachteten Werte für Quecksilberdampf. Diese Berechnungen wurden unter der Annahme durchgeführt, daß der Kondensationskoeffizient gleich Eins ist. Es zeigt sich, daß die Nusselt-Zahl parametrisch vom Sättigungsdruck und von der Länge der kondensierenden Oberfläche abhängt. Die berechnete Massen-Kondensationsrate ist geringer, als sich aus einer das Temperaturgefälle an der Grenzschicht nicht berücksichtigenden Theorie ergibt. Die Resultate sind alle anwendbar für ausreichend lange Platten bei nicht zu geringem Druck. Bei kurzen Platten und Drucken unter 25 mm Hg müssen weitere Terme der asymptotischen Reihenentwicklung berücksichtigt werden.

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Hyperelastic Multiphase Porous Media with Strain-Dependent Retention Laws

This article presents poroelastic laws accounting for a retention behavior dependent also on porosity, as suggested by experimental evidence. Motivated by the numerical formulation of the corresponding boundary-value problem presented in a companion article, these constitutive equations employ displacements and fluid pressures as primary variables. The thermodynamic admissibility of the proposed rate laws for stress and fluid contents is assessed by means of symmetry and Maxwell conditions obtained from the Biot theory. In the case of strain-dependent saturation, the two elasticity tensors describing the drained response in saturated and unsaturated conditions, respectively, are proven to be in general not coincident, with their difference depending on capillary pressure and porosity. Furthermore, it is shown that besides the stress decomposition proposed by Coussy, also the stress split proposed by Lewis and Schrefler is consistent with the Biot framework. The former decomposition is obtained for retention laws depending only on capillary pressure, as expected. The Lewis–Schrefler split is proven to be consistent with retention models depending also on porosity. In these developments, the compressibility of all the phases is taken into account, in order to assess the thermodynamic consistency of an extension of the Biot’s coefficient to partially saturated anisotropic porous media.