A thermodynamic approach with internal variables using Lagrange formalism. Part I: General framework

We present some reflections on the application of the Lagrangian formalism for continuous media locally uniform subjected to internal irreversible evolutions. The Lagrangian density, defined as the time derivative of a non-equilibrium thermodynamic potential, [Thermodynamics of Relaxation Processes using Internal variables within a Lagrange-formalism. P. Germain’s Anniversary Volume 2000. Contiuum Thermomechanics: the Art and Science of Modeling Matter’s Behaviour, 2000], contains all the symmetry properties of the system. The generalised Lagrange co-ordinates correspond to the state and internal variables of the time derivative of the generalised Gibbs potential. The latter being used within the framework of the De Donder’s method, must also account for the memory effect of the physical medium.This first part is devoted to the thermodynamic framework called the distribution of non-linear relaxations approach (DNLR) developed by C. Cunat on the basis of the generalised Gibbs’ relation.     

A thermodynamic approach with internal variables using Lagrange formalism. Part II. Continuous symmetries in the case of the time–temperature equivalence

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium.     

Etude par effusiometrie de torsion des proprietes thermodynamiques en phase solide des laitons binaries (Cu,Zn)

The Gibbs thermodynamic variables which we have determined for thirteen brass alloys (Cu, Zn) are in good accord with the literature. The experimental conditions were as close as possible to those of sublimation in static vacuum. The application of the test of the third law in β-brass shows that our free enthalphy measurements are in agreement with the measurements of specific heat by Moser. The primary solid solution α copper of the binary brasses shows quasi-regular behaviour over the whole composition range. Our measurements seem to indicate the existence of a β′/(α + γ) eutectoïd at a temperature less than 100°C. The high degree of order of the γ phase is seen in the large negative values of excess entropy.     

Modelling of Progressive Strain Tests by a Non-linear Relaxation Approach

A model based on the thermodynamics of relaxations of continuous media (called DNLR as Distribution of nonlinear relaxations) is used to describe the tension/torsion progressive strain phenomenon. The proposed model is compared to the experimental data obtained by Benham on mild steel at room temperature with satisfactory results. We discuss the capacities and the relevance of the model obtained.     

Optical techniques for in situ dynamical investigation of plastic damage

We follow the damage process of high-density polyethylene during tensile tests. We simultaneously track changes in the density and average orientation of cavities using incoherent light transport. At the same time, we measure the true strain with a video-extensometer and the heat with an infrared imager. We see that the damage process has two major separate steps. First, a globally isotropic nucleation and growth of cavities occurs up to a deformation of about 1.1. Then, at higher deformations, cavities stop growing. Instead, they progressively orient and elongate along the tensile axis. The transition between these two damage processes seems to be related to strong physical and geometrical constraints, also probed through a typical thermal signature.     

Rheological constitutive equation of solids: a link between models based on irreversible thermodynamics and on fractional order derivative equations

In this paper we focus on the rheological problem of defining a constitutive equation for viscoelastic materials. In this simple case, we show that writing the dissipative component of the observable response to a given excitation as the result of multiple internal processes working for equilibrium recovery (flux of internal hidden variables), can yield a recursive series in time. This can be obtained when use is made of the theorem of created entropy equipartition as a model for fluctuation regression. A distribution (spectrum) for relaxation times naturally follows. The model thus obtained reflects the concept of a hierarchically constrained dynamic behavior. The conclusion is that the operator of non-integer differentiation in time applied to field variables can also be recovered from pure thermodynamic considerations.     

Continuous symmetries and constitutive laws of dissipative materials within a thermodynamic framework of relaxation: Part I: Formal aspects

An analysis of the continuous symmetries of the constitutive laws of inelastic materials written within a thermodynamical framework of relaxation is performed. This framework relies on the generalization of Gibb’s relationship outside the equilibrium of a uniform system, and the use of the fluctuation theory to model the material dissipation due to its internal microstructure change [Cunat, C., 2001. The DNLR approach and relaxation phenomena. Part I – Historical account and DNLR formalism. Mech. Time-depend. Mater. 5, 39–65]. The approach leads to a viscoelastic like formulation for small deformations, and changes gradually for finite strains towards elastoviscoplasticity (with or without damage) via a dependence of characteristic times with the loading path, in a way similar to the endochronic approach developed by Valanis [Valanis, K.C., 1975. On the fundations of the endochronic theory of viscoplasticity. Arch. Mech. 27, 857–868]. The present thermodynamic framework has been previously applied to elastoviscoplastic materials under cyclic and non-proportional loadings [Dieng, L., Abdul-Latif, A., Haboussi, M., Cunat, C., 2005b. Cyclic plasticity modeling with the distribution of non-linear relaxations approach. Int. J. Plasticity 21, 353–379]. The constitutive laws split into the state laws relating intensive variables (thermodynamics forces) to extensive-like variables, and the complementary evolution laws of the internal variables associated to the dissipative mechanisms. An interpretation of a non-equilibrium thermodynamic approach of irreversible processes in terms of an extremum principle is proposed, associated to a Lagrangian functional. It is shown that one possible choice for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term that accounts for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler–Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is formulated. It is further shown that both symmetry conditions are fully equivalent along the optimal path corresponding to the satisfaction of the constitutive laws. In terms of both practical and methodological aspects, the predictive nature of the symmetry analysis is highlighted, as a systematic tool for the exploitation of the constitutive response. Its performance and utility are exemplified by the construction of a time–temperature equivalence principle for a dry viscous polymer (PA66); the calculated shift factor is shown to well agree with the empirical shift factor given by Williams–Landel–Ferry (WLF) expression. A systematic interpretation of the calculated symmetry groups of the constitutive laws in terms of master curves for various plastic and viscoplastic materials shall be presented in a forthcoming contribution.     

Determination electrochimique,par pile a electrolyte solide CaF2, des proprietes thermodynamiques de formation des alliages binaires (Ca,Cu) entre 800 et 1300 K

Thermodynamic quantities of formation of the CaCu system intermetallic phases are, for the first time, directly determined by means of electromotive force measurements. The alloy sample is prepared, in situ, on a copper point electrode by electrolysis. The activity of calcium in CaCu alloys is directly compared with that of pure calcium (liquid or solid) produced in the same conditions on an inert (iron) electrode.     

Formulation DNLR intégrale des lois de comportement : généralisation du principe de superposition de BoltzmannDNLR integral formulation of behaviour laws: generalisation of Boltzmann's superposition principle.

A thermodynamic approach of relaxation phenomena called DNLR (Distribution of Non Linear Relaxations) has been employed for about fifteen years to describe the behaviour of materials. In this Note, we show that different DNLR constitutive equations, initially written in a rate formulation, can be put in a time-integrated (hereditary) form which generalises the superposition principle to non linear fields like viscoplasticity or damage behaviours. To cite this article: B. Martin et al., C. R. Mecanique 332 (2004).