Elsevier

Mechanics of Materials

Volume 80, Part A, January 2015, Pages 113-123
Mechanics of Materials

Determination of dissipated energy fields from temperature mappings on a rubber-like structural sample: Experiments and comparison to numerical simulations

https://doi.org/10.1016/j.mechmat.2014.09.010Get rights and content

Abstract

The main goal of this paper is to provide an experimental way to deduce the field of dissipated energy from the measurement of the temperature field. The case studied is a structural sample (i.e. with a circular notch) in order to test the ability of the protocol to describe heterogeneous energy fields. The material chosen is a filled synthetic elastomer in order to investigate the case of large displacements. Moreover, very thin samples are used to reduce the solving of the thermo-mechanical problem to a 2D investigation. To reach high enough spatial and thermal resolutions the protocol suggested takes advantage of the low thermal conductivity of the material and of a very precise calibration of the infrared camera used to record the temperature fields. The paper suggests a protocol based on an adiabatic assumption and investigates its capabilities and range of validity. A very accurate description of the severe gradients of the fields of dissipated energy can be obtained. In order to evaluate the reliability of the experimental fields, a numerical approach is applied using a simple yet robust dissipation modelling. Finally, the experimental fields of displacement and of dissipated energy are compared with the ones obtained from the numerical simulation.

Introduction

The temperature evolution of materials subjected to mechanical loading is a phenomenon highlighted for long (Gough, 1805). This evolution is related to internal heat sources of various natures and to both time and spatial evolutions induced by conduction inside the material and heat exchange at the boundaries. Considering temperature as itself is already a precious tool to characterize the response of materials or structures under mechanical loading. This analysis can be performed for homogeneous cases under monotonic or cyclic solicitations in order to include any thermal effect induced by the mechanical testing conditions (heat build-up for fatigue testing, for example), to investigate the thermo-elastic couplings (Delpueyo et al., 2012), to follow the evolution of the material microstructure (Anthony et al., 1942, Le Saux et al., 2012, Delpueyo et al., 2012) or to detect a change in the thermo-mechanical response, allowing for example to get a quick evaluation of the fatigue limit (La Rosa and Risitano, 2000, Le Saux et al., 2010). Taking advantage of the development of full field measurements techniques (and especially of infrared devices), the analysis of heterogeneous cases is also available. A first way to use them is to spot a localized phenomena for inspection or monitoring purposes, to assess the location of fatigue damage or to detect the occurrence of a localized phenomena at the macroscopic or microscopic scale (Marco et al., 2013, Chrysochoos and Louche, 2000). A second way is to use the full temperature field to evaluate the heterogeneous mechanical field, under the hypothesis of thermo-elasticity (Bodelot et al., 2009), adiabaticity (Kapoor and Nemat-Nasser, 1998) or by a comparison to finite element computation (Ranc et al., 2014).

As evoked previously, the temperature evolution is not intrinsic to the material nor to the structure behaviour and depends on the tested volume and geometry, the thermal boundary conditions and the ratio between the thermal and mechanical characteristic times. Moreover, several heat sources may be activated by the mechanical loading and temperature is only a global indicator of this combination (Chrysochoos, 2012). The evaluation of these heat sources is clearly a key point in order to characterize and understand the thermodynamic behaviour of the sample tested. The applications lead for instance to evaluate the balance between stored and dissipated energy (Chrysochoos, 2012, Rittel, 1999), to identify the intrinsic thermo-mechanical parameters (Lemaitre and Chaboche, 1985), to detect localized phenomena (Chrysochoos et al., 2009, Wang et al., 2014) or microstructural changes (Delpueyo et al., 2012, Samaca Martinez et al., 2013) and to feed energy based criteria to predict the fatigue properties (Doudard et al., 2005, Le Saux et al., 2010).

The reconstruction of heat sources mappings from localized or 2D thermal measurements is performed by solving the thermo-mechanical problem. This is usually not an easy task as this problem is time and space dependant. Moreover, it could combine mechanical and thermal fields that are both heterogeneous, that can stabilize or evolve along time and that are possibly lightly or strongly coupled. The approaches used to identify the heat sources are therefore combining experimental and numerical aspects, with a ratio of these aspects depending on time, space and phenomena complexities. All strategies take advantage of any assumptions relevant to simplify the resolution of the thermo-mechanical problem. A first consideration deals with the ratio between the characteristic thermal times (for conduction and exchange) and the mechanical one. For example, high speed tests could profit from an adiabatic analysis (Kapoor and Nemat-Nasser, 1998), while quasi static test may go for an isothermal assumption (Stainier and Ortiz, 2010). A second important consideration is the hypothesis of a transient or stabilized thermodynamic state of the sample considered. For example, stabilized cyclic tests could take advantage of the measurements of both cyclic instantaneous thermal response and of mean temperature evolution to split the contribution of thermo-elastic couplings from the one of the intrinsic dissipation (Chrysochoos and Louche, 2000, Chrysochoos, 2012), while monotonic quasi-static tests lead to an evolution of all thermodynamic variables and induce a more complex identification. The last main consideration deals with spatial hypothesis and a first classical analysis is the so-called “0 dimension” approach which consists in performing a mean operation over the gauge volume of the sample, assuming that the heat sources are homogeneous within the volume. The resolution of the heat equation could then become analytical and the link between the temperature and the heat sources is direct. When this assumption is not justified (i.e. in most cases), the resolution of the problem requires additional information or hypothesis on the heat source distribution. Two main approaches can be found in the literature with or without the additional input coming from the mechanical data.

The first approach is a “direct” approach, reconstructing the heat sources from the temperature measured. A first resolution technique aims at solving numerically the heat equation by using regularizing techniques or special local least square fitting of the experimental thermal signal (Chrysochoos et al., 2009) in order to be able to evaluate the differential term of the heat equation, even for a 2D case (Wang et al., 2014). A second technique consists in using Fourier techniques to suggest a spectral solution of the heat equation. This spectral basis can be composed of eigen functions that are compatible with the boundary conditions (Chrysochoos and Louche, 2000), or defined more freely, by adding an extra function to describe the boundary conditions before rebuilding an orthogonal basis (Doudard et al., 2010). The main drawback here is the smoothing operations to evaluate the differential operators, which impacts the accuracy and the spatial resolution of the heat sources (Chrysochoos and Louche, 2000).

The second main approach used to evaluate the heat sources is to perform an “inverse” analysis where the distribution of the heat sources is assumed and the magnitude is adjusted from the comparison between experimental and computed thermal fields on the basis of an analytical (Doudard et al., 2010) or a finite elements simulation (Ranc et al., 2014, Le Saux et al., 2013). This second option remains difficult for numerous reasons: the thermomechanical problem is usually ill posed and can lead to several solutions, the thermal exchange conditions are difficult to evaluate, especially when large displacements occur and the conversion of mechanical energy into dissipated energy is often complex with numerous parameters.

In this paper, an elastomeric material is tested. The thermo-mechanical characterization of these materials is challenging because energetic and entropic contributions are both at work, with possible changes of the microstructure (crystallisation, fillers network…). Moreover, the displacements involved are usually very large, making difficult to follow the same material configuration (Pottier et al., 2009, Le Chenadec et al., 2009). Another difficulty for these organic materials comes from the thermal dependency of the constitutive response, leading to a coupling when the temperature increases. The last but not least complexity comes from the multiplicity of the phenomena involved in the hysteretic loop which makes difficult the identification of the energy balance (Le Saux et al., 2010, Medalia, 2013, Samaca Martinez et al., 2013). From an industrial point of view and especially for anti-vibrations applications, this topic is clearly important because of two issues. The first one comes from the good damping properties (and therefore the strong conversion into heat) and the low stiffness (requiring massive components) of these materials, coupled to a very low thermal conductivity. This combination leads to an inhomogeneous temperature rise of the rubber component under cyclic loadings called heat build-up in the literature (Medalia, 2013), and that strongly influences the mechanical response, the fatigue properties and can even lead to thermal ageing. Another main topic is the use of the dissipated energy as an efficient fatigue criterion (Le Saux et al., 2010), that is crucial to design these components.

The aim of this paper is to present a specific procedure to evaluate the heat sources mappings directly from the experimental measurements. The paper is divided into four sections. In the first one, some information about the material and the experimental devices used are described. The idea is to take advantage of the low heat conductivity of the rubber to test the ability of an adiabatic analysis to capture the gradients of the heat sources. The focus is given here on gradients in 2 dimensions and a very thin sample has been chosen so that 2D surface measurements can be representative of the whole volume. The sample geometry is classical (a plate with a hole) in order to induce strong gradients but allowing for a classical numerical analysis. The solicitation is cyclic, in order to study a given mechanical configuration, which permits to follow the same material configuration without any reconstruction (Pottier et al., 2009) and to separate the thermo-elastic coupling contributions from the intrinsic dissipation. To reach high enough spatial and thermal resolutions the proposed protocol takes advantage of the low thermal conductivity of the material and of a very precise calibration of the infrared camera used to record the temperature fields. In the second section, the thermodynamical theoretical framework is given and a specific experimental protocol is tested and discussed to evaluate the heat sources mappings from the temperature measurements. In order to evaluate the consistency of the energy fields obtained, a comparison to a numerical simulation is aimed at. In the third section, the modelling and numerical tools used to achieve a finite elements simulation of the test are presented. Finally, the experimental and numerical fields of dissipated energy are compared and exhibit a very good correlation.

Section snippets

Material and specimen

In this study, a synthetic rubber filled with carbon blacks is considered. This non-crystallizing compound is classically used for automotive components. Table 1 gives some information on the material recipe and Table 2 the main thermal properties. Fig. 1 illustrates the constitutive response of this compound, for a cyclic tension test achieved on a classical H2 sample. The classical features of a filled elastomeric material can be highlighted here: the mechanical response exhibits a

Theoretical framework

The thermodynamics of irreversible processes is used here to analyze the thermal results. The thermodynamical equilibrium state of a volume element is defined at each instant t by the current value of a set of variables, namely T the temperature, ε, a strain tensor and Vk the internal variables. Internal variables are chosen in accordance with the physical mechanisms within the studied material. The first and second principles of thermodynamics are then checked in order to define the evolution

Numerical simulations

In this section, the modeling and numerical tools used to simulate the experiments and to evaluate the mechanical fields are presented. It is worth underlining that the goal here is to predict the fields of dissipated energy and not the temperature fields. This leads of course to a much simple framework as the thermal parameters (material physical data, boundary conditions) are not needed.

Comparison between experimental and numerical results

In this section the results obtained from the simulations are compared to the experimental data. The relevancy of the modelling is tested on the correlation between the real and simulated geometries of the sample and on the force–displacement curves. Finally, the energy fields obtained experimentally and by simulation are compared and discussed.

Conclusion

Measuring the actual dissipated energy is always a difficult task, even more when the thermo-mechanical case considered is heterogeneous. In this study, the goal was to evaluate the heat source terms without the help of any numerical simulations and a protocol was suggested, tested and discussed. This protocol, based on the initial temperature variation clearly offers a very good opportunity to capture accurately the dissipation fields, even when severe gradients are involved. The application

Acknowledgements

The authors would like to thank the ANR for its financial support (ANR-2010-RMNP-010-01) and all the partners of the PROFEM project: TrelleborgVibracoustic, GeM, UBS and LRCCP. Helpful discussions with Dr. Cédric Doudard from ENSTA Bretagne are also acknowledged.

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