Christian Cunat

The DNLR approach and relaxation phenomena. Part I - Historical account and DNLR formalism

The relaxation phenomena defined by De Groot and Mazur (1962)describe the internal reorganizations linked to the return toequilibrium of media subjected to external perturbations of lowamplitude (near the equilibrium state). Far from equilibrium, anytheoretical approach to these phenomena has to include the followinginformation: the internal reorganizations are multiple and theirkinetics are nonlinear. Indeed, much experimental evidence has lead tothis conclusion. A classical example for the analysis of relaxationsnear the glass transition is the experimental study of the volumerecovery of PVAc (Polyvinylacetate) done by Kovacs (1963).Over many years, we have developed an approach in the framework ofirreversible thermodynamics, called the Distribution of Non-LinearRelaxations (DNLR) to establish constitutive laws for various materialsunder coupled physical solicitations. It is based on a generalization ofthe fundamental Gibbs equation (1902) for systems outside equilibrium.This relation combines the two laws of thermodynamics into a singleexpression; for example, the internal energy e =e(s, v, ni, ¨)depends on the whole of the state variables, including theentropy s. The salient points of the DNLR approach are (i)to naturally take account of the couplings found in physics, (ii) themultiplicity of the mechanisms of internal reorganization and (iii) thenonlinearity of the kinetics for the return to equilibrium.The aim of this paper is then (i) to present in this first part thebases, the formalism, and the framework of the DNLR approach and (ii) ina second part to check the pertinence of this general DNLR strategy tosimulate the experimental data of Kovacs concerning PVAc. This developedmodeling will be compared to other works already done in the literature.

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

Abstract 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.

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.

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

Abstract 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.

In situ mechanical characterization of polymers with the association of three optical techniques

A combination of three optical techniques is presented for in situ monitoring of macroscopic and microscopic variables characterizing the deformation of polymers. A video extensometer allows for the monitoring of true stress and strain. An infrared imager along with an appropriate mathematical algorithm allows for the determination of the energies that are converted into heat during the whole test. This makes possible a quantification of thermomechanical couplings revealing structural effects at microscopic scales. Finally, incoherent steady light transport is used to produce images of backscattered intensities. A physical model enables the authors to follow damaging processes occurring at microscopic scales.

Modeling of the Mechanical Behavior of a Nickel Alloy by Using a Time-Dependent Thermodynamic Approach to Relaxations of Continuous Media

In this paper, we show the capabilities of a thermodynamicapproach to the relaxation of continuous media to describe the behaviorof the Nickel-based alloy IN738LC submitted to various mechanicalloadings at high temperature (850°C). A preliminarysimple modeling (with one spectrum of relaxation times) does not providesatisfactory results. Microscopic observations of the alloy (Sievert etal., 1997) show the existence of several viscosity domains and we takeone of them into account in our modeling of two spectra of relaxationtimes. The results are much better. Finally, we compare our simulationsto those obtained with the classical elastoviscoplastic modeling ofChaboche and we link these two approaches at the conceptual level.

SECTION 9.17 – Physical Aging and Glass Transition of Polymers

This chapter discusses the physical aging and glass transition of polymers. In the case of glassy materials near the glass transition, the aging transformation is usually associated with the recovery of a thermodynamic equilibrium state. It concerns various physical properties, such as the mechanical compliance, the refractive index, the apparent volume expansion, and the enthalpy related to the apparent specific heat. This chapter presents a theoretical modeling that is based on an irreversible thermodynamics approach called “the distribution of nonlinear relaxations” (DNLR). Further, two versions (called versions 1 and 2) corresponding to different levels of approximation of the DNLR formalism are developed to describe the volume recovery under various thermal histories. Version 1 discusses approximation without entropic coupling and version 2 discusses approximation with entropic coupling.

Visible, Infrared and Incoherent Light Transport Optical Techniques for in Situ Material Testing

Three experimental non intrusive techniques are used simultaneously for the metrological monitoring of the deformation of a material in a tensile load machine. The first technique is a video-extensometer (Videotraction®) based on ink spots tracing. The image treatment is fast enough to allow an active control of the machine. It furnishes the true stress-true strain curves for a given specimen according to a desired loading path. An infrared camera is placed on the opposite side of the sample and records some picture of the temperature field evolution during the whole experiment. Thanks to a numerical technique developed for solving the inverse problem of heat source reconstruction, it is possible to get the dissipated power versus strain curve. This brings a new information, of thermodynamic nature, reflecting the thermal activity of the internal mechanisms of deformation. Finally, on the same side of the sample, a third optical technique is used. This technique applies to turbid samples (neither absorbent nor completely transparent to visible light). A laser impact the sample on a very small spot (50 μm). An incoherent steady-light transport (ISLT) occurs within the turbid sample. Thanks to a high resolution camera, the backward scattering image (10 mm) is analysed through a theoretical modelling of incoherent light transport in disordered materials. It allows to follow the concentration of the scattering objects as well as their anisotropy during the experiment thus yielding additional information about what happens in the microstructure of the sample during the test.

From Thermomechanical Heat Source Reconstruction to the Validation of Mechanical Behavior’s Laws

The development of thermal imaging in the past 25 years lead the mechanical sciences community to use this tool in order to analyze the thermal effects accompanying deformation processes of materials. At first, it allows to check some qualitative effects observed previously when thermocouples were embedded in the specimens. The discussions were relying on a qualitative analysis of the temperature behavior associated to each thermomechanical behavior (thermoelasticity, viscoplasticity, damage...). Then it appears clearly that some source reconstruction was unavoidable as temperature is not an intrinsic variable to describe the internal thermal processes: it depends on heat exchanges with the surroundings i.e. the external dissipation or entropy flux to use the appropriate language of thermodynamics of irreversible processes. Until now, very limited observed thermomechanical behaviors have been properly reproduced with a consequentially model. But some attempts have been done that illustrates all the benefits ones could retrieve from such achievement [Chrysochoos, 1992,2000]