Application des réseaux de Bragg à la détermination des déformations distribuées dans les matériaux composites

The behaviour of brittle matrix fibre reinforced-composites is dependent of the properties of its constituents (fibre, matrix and fibre-matrix interface) and is affected by the mechanical and environmental loads. The fracture of a unidirectional composite material is the result of various micro-mechanisms (matrix and/or fiber fractures, crack fibre interactions, interface failure, delamination, …) that generally occur simultaneously. Among these, one distinguishes the failure by fibre-matrix debonding and the frictional sliding along the composite interfaces. The initiation and debonding growth are influenced by the specimens' geometry, the loading conditions, the nature of the constituent materials, the residual stress and the fibre-matrix interface. Several works have been carried out that deal with theoretical and experimental aspects of micro-mechanisms of fracture. However, a complete understanding is still lacking due to the difficulties to characterise quantitatively the individual contribution of these different micro-mechanisms. In the recent past, optical Fibre Bragg Grating (FBG) have been used as embedded strain sensors. They are particularly adapted to polymer composites where they can provide accurate internal strain measurements at selected locations. Until today, short FBGs have been often used in uniform loading conditions. In such cases, the grating parameters do not vary along the grating length and when uniform variations in strain and/or temperature occur, the FBG spectral response exhibits a simple shift of the Bragg peak without modification of the spectrum shape. This wavelength shift has been used extensively as a global indicator of internal strain in classical sensing applications. This approach however is not realistic when the FBG is found near damage, i.e. crack, delamination, residual strain field, etc. To achieve non-uniform strain measurements using long gauge FBG sensors, a new Optical Low-Coherence Reflectometry (OLCR-based) technique has been developed and tested in order to study crack-fiber interaction and the interface between the glass fibre and epoxy matrix material during composite damage process. When a long FBGs is subjected to non-homogeneous distributed strains, the grating parameters are position-dependent. Due to the extensive grating length, the optical sensor undergoes substantial non-uniform variation along the fibre direction. Inhomogeneity of the grating is a result of the application of a non-uniform strain or temperature field in the interface region and/or the FBG writing process (manufacture). The coupled-mode formalism provides a mathematical tool to describe the interaction of light propagation. This analysis leads to the introduction of a unique complex coupling coefficient to be determined from the FBG complex impulse response. A novel optical low coherence reflectometry acquisition system, designed at EPFL, allows precise measurement of the FBG complex impulse response with high precision and low noise (generally below -120dB). The working principle of the apparatus is well documented elsewhere (93). The OLCR-based method enabled us to obtain the local Bragg wavelength along the grating located in the core of the optical fibre. In this study, epoxy is used as the main material. An established standard protocol is followed with a couple of D.E.R. 330/732 Dow Epoxy Resins (70%/30% mass weight) and a D.E.H. 26 curing agent (+13%). Once the polymerization process is well advanced (24h@28°C), the specimen is removed from its mould and placed in an oven for the post curing phase (10h@70°C). The specimen is left to cool inside the oven until ambient temperature was reached. Three main experiments are conducted with a centrally located long FBG (14 to 25mm length): one on a Double Edge Notch Specimen (DENS) in order to investigate and validate the newly-developed OLCR method and two pull-out tests on two different single fibre cylindrical specimens (with or without fibre protective acrylate coating). The last experiments are conducted in order to describe the interface behavior. Thus, the optical fibre plays the role of fibre-reinforcement and strain sensor. Various Finite Element Models are developed in order to understand better the phenomena and analyse the experimental results. All materials are considered to be linear elastic and with isotropic properties. To model the matrix shrinkage effect the problem is considered analogous to a thermo-elastic one. A shrinkage function, evaluated experimentally, from the FBG response is introduced in the general strain-stress relations. Displacement boundary conditions are applied to the fibre end and the epoxy to simulate the loading conditions. The first set of simulations is dedicated to the stress analysis of the DENS and the comparison of the OLCR data with those of the simulations. The other set allows to identify the properties of the fiber/matrix for weak (glass-acrylate) or strong interfaces (acrylate-epoxy or glass-epoxy) respectively. In particular for the weak interface, interaction between the glass fibre and the coating is described using the regularized Coulomb friction model. That defined the critical shear stress as a fraction of the contact pressure. Higher than this limit, the contact surfaces start sliding relative to one another. Except of the friction coefficient μ, an elastic slip parameter γlim is also introduced into the model to take into account the interface stiffness in the elastic regime (i.e. the sticking stiffness). For the strong interface, interaction between the glass fibre and the epoxy resin is described using the energy release rate G parameter. The observed interface crack grow is used as an input to simulate evolution during debonding propagation. The J-integral equivalent is numerically calculated around the crack tip for different crack lengths during the test. A G(z) distribution along the glass-epoxy interface is identified as possible criterion for adhesion rupture. A single Coulomb identification model μ is proposed to describe the behavior of the interface once it fails. The main results of the simulations are given bellow: The redistribution of the axial stress is obtained for two interfaces and at different force levels. The strain data show that residual stress effects due to epoxy matrix shrinkage and positive strains reveal the presence of the interfacial crack propagation when the fibre is in tension. Numerical models are compared with experimental results of the fibre pull-out tests. The overall experimental results are reproduced for the proposed parameter values although discrepancies exist when the load is released. Two parameter models explain the data very well. At the glass-acrylate weak interface: two interface parameters μ and γlim are identified. At the glass-epoxy strong interface: a constant friction coefficient μ and a distributed energy release rate G(z) are estimated. In this study, a mixed experimental-numerical approach is developed to investigate the strain response of long-gauge FBG and the interface of glass-epoxy and glass-acrylate epoxy. This work demonstrated the potential of the FBG sensor associated with OLCR technique to understand micromechanics damage in fracture.

Assessment of mechanical properties in unirradiated and irradiated zircaloys and steels with non-standard tests and finite element calculations

Emiliano Campitelli

Nuclear materials have been successfully employed in commercial fission reactors for many decades. Research and material development for advanced fission system and future thermo-nuclear fusion reactors have acquired maturity while great challenges remain to be taken up. The goal of the material scientist is to understand the way neutron irradiation and aggressive reactor environment alter the original material properties and to safely manage the operation conditions for each particular reactor component. In the framework of surveillance programme, the need has arisen to design experimental methods to be used with a limited amount of material for Post Irradiation Examination (PIE) of reactor components to test the integrity of those on a regular time basis. Having to deal with radioactive material, it is particularly important to devise experimental testing procedures as simple as possible but from which reliable information can be drawn. Within materials development programme, irradiations have continuously been performed either in reactors or with particle accelerators. In both cases, the irradiation volumes are usually limited. The concern was to optimize the volume of the irradiated samples in order to increase their number to obtain a good statistical representation of their mechanical behavior after irradiation, and to have a very definite and homogeneous damage dose and irradiation temperature. The concerns explained above originated the development of small-scale specimen testing methods. Advantage is taken of the fact that, within size and geometry limitations, some mechanical properties of a sample are independent of its absolute size. In this way it is possible to perform mechanical tests on samples of reduced volume that represents big advantages to face the aforementioned challenges. Non-standard test on very small specimen have also been proposed to extract mechanical properties of the bulk. If these non-standard testing methods are to be employed for PIE, it becomes necessary to develop modeling tool to understand under what circumstances they can provide useful information and specially how to obtain it. Finite element modeling (FEM) of the non-standard tests has played a key role in this sense, providing a link between standard test analysis and that of the non-standard ones. The goal of this work was to implement finite element models for non-standard tests to ultimately study the mechanical properties of Zircaloy cladding tube and tempered martensitic steels in the unirradiated and irradiated condition. One model was developed for ring tensile tests. It was first validated with a quite ductile iron-chromium ferritic alloy, for which ring and uniaxial tensile test were performed at room temperature. Using the constitutive behavior determined from the uniaxial tensile tests, the load – displacement curves of the ring tensile tests were very well reproduced. Another model was developed for the so-called small ball punch test, which consists in deforming a 3 mm diameter disk clamped between two dies with a ball puncher and to record the load – displacement curve. In order to validate the model, a series of ball punch tests and of tensile tests were carried out on the austenitic steel 316L at room temperature. In this last case, the ball punch test curves were also well reconstructed with the model. Based upon a simple analysis and simulation results, we showed that it is possible to obtain, by ring tensile tests, a quick and accurate assessment of the constitutive behavior of an unknown material, simply by processing the experimental force-elongation data in the same way as for uniaxial tests, taking half the ring perimeter as initial length and twice the ring cross section as the sample cross section. The ring tensile test method was used to determine the hoop mechanical properties of Zircaloy tubes used in fission reactors. Tensile tests were also performed in the axial direction of the Zircaloy tubes. Differences between the yield stress as well as the post-yield behaviors in the axial direction and those in the circumferential direction of the tube could be measured reflecting some component of the anisotropy of the tube. Ball punch specimens were cut also out of the tube in the radial direction and tested. Using the FE model for punch test, in which the anisotropy of the mechanical properties are taken into account through the Hill's theory, the values of the Hill's yield function parameters were estimated. The irradiation hardening of neutron irradiated Zircaloy tube was finally determined. The ball punch test model was extensively used for numerical investigations, to determine how the constitutive behavior of the materials (uniaxial true stress – true strain) employed as input acts in combination with other experimental factors (specimen thickness, friction coefficient) in mediating the shape of the experimental curves. In the light of the results, we derived calibrations between parameters of the non-standard and standard tests. For instance we proposed a new approach to determine the yield stress from the ball punch test. We also showed how the slope of the punch tests scales with the average flow stress of the material. The calibrations proposed in the past by authors and published in the open literature have been revisited and critically reviewed. Based upon the findings of the numerical studies, we propose a given methodology to unambiguously evaluate the material constitutive behavior of a material in a case where the only available data comes from various non-standard tests. Experimental ball punch test results obtained on proton-irradiated specimen (0.5 dpa and Tirr = 523 K), of the tempered martensitic steel, EUROFER97, have been obtained. The irradiation hardening was found equal to about 150 MPa, consistently with previous data on similar steels and irradiation conditions. From the simulations of the ball punch test curves of the irradiated specimens, it was concluded that the irradiation does not affect the strain hardening at relatively low fluence.

EPFL2005

The response of embedded FBG sensors to non-uniform strains in CFRP composites during processing and delamination

Larissa Sorensen

From airplanes to sailboats to bridges, composite materials have become a significant part of our everyday structures. With increasing demand, these materials are pushed to their limits to improve structural efficiency. As a consequence, research and development must continually improve products and provide support for the end user who will need to know the characteristics of their new material. Progress made in the area of optical fibre sensing has opened new avenues for measuring and monitoring fibre-reinforced polymer (FRP) composites, since they can be embedded directly into the composite during manufacturing. These globally noninvasive sensors can provide internal strain and temperature measurements from the moment processing starts until the final failure of the part. The goal of this research is to develop and demonstrate fibre optic sensing techniques that can characterize the internal strain state of FRP composites. In particular, this work focuses on measuring three-dimensional, non-uniform strain fields in carbon fibre-reinforced polymers (CFRP) using fibre Bragg grating (FBG) sensors. Although FBG sensors are becoming widespread for simple uniaxial strain measurements, their response to complex, non-homogeneous strain fields is still difficult to interpret. To illustrate advances in both experimental techniques and the interpretation of measured FBG data, two main areas of composite monitoring are addressed. They include the study of residual strain evolution and of delamination cracking, which both produce non-homogeneous strain fields. Unidirectional carbon fibre-reinforced polyphenylene sulphide (AS4/PPS) laminates are observed during processing to measure residual strain progression, and then later subjected to Mode I double cantilever beam delamination tests. These thermoplastic composite specimens are also produced in a cross-ply configuration, for the purpose of residual strain monitoring. In each laminate, a long-gauge length (20-35 mm) FBG is embedded parallel to the reinforcing fibres, and centred along the length of the plate. Results of polarization sensitive FBG monitoring indicate characteristic material state changes such as the glass-transition and the melting temperatures. These measurements take advantage of both the transverse and longitudinal strain sensitivity of the FBG. When transverse strains are unequal they induce birefringence in the FBG (defining a fast and a slow axis), which results in a split of the normally bell-shaped reflected spectrum. An evolution of this birefringence is monitored during cooling, culminating in average residual transverse strain differences in the embedded FBGs of 230 με and 410 με for unidirectional and cross-ply specimens respectively. Based on the wavelengths measured along the fast polarization axis of the fibre, (observed to be less sensitive to transverse strains) cross-ply specimens exhibit absolute longitudinal residual strains in the order of -350 με. Small longitudinal strain values are the result of the low coefficient of thermal expansion of the carbon reinforcing fibres. An important step forward in FBG monitoring is taken by measuring the absolute values of the three unequal principal strains in a composite material without making assumptions about the state of strain in the FBG (i.e. diametric loads, plane stress, axisymmetry, etc.). For this purpose, a polarization controlled, hybrid FBG-Fabry Pérot optical sensing technique is developed to measure residual strain evolution. The Fabry Pérot sensor used in this hybrid method is only sensitive to longitudinal strains, thus providing the additional data required to solve the three-dimensional strain state directly. To better understand the state of residual strain in the composite material, a temperature dependent thermoelastic finite element model is employed to investigate the strain accumulation during cooling. By comparing modelled results to the data from the optical fibre, it is shown that the mould influences the residual strain development during cooling, and that some of these strains are released after demoulding. Examination of the simulated and experimental curves indicates that the final residual strain state observed with the FBG is close to that of a freely cooling composite plate. Since the embedded optical fibre is a local inclusion, its strain state is not necessarily that of its host material. In this work, finite element models are used to determine the stresses and strains developed in the surrounding composite material. Near the optical fibre, there is a perturbation of the strain field that extends to a distance of three fibre diameters. In the far-field of cross-ply specimens, tensile transverse stresses reach half the matrix fracture strength. This may help to explain matrix cracking observed on the surface of these specimens. The second portion of this study is aimed at the measurement of non-uniform longitudinal strains superimposed on an already three-dimensional residual strain state. A polarization adapted optical low coherence reflectometry (OLCR) technique takes distributed measurements of the local Bragg wavelengths for a given polarization axis. For a constant state of birefringence, one can relate the distributed wavelengths to the non-uniform longitudinal strains along the length of the FBG sensor. Delamination cracking in double cantilever beam specimens creates a non-uniform strain field ideally suited to illustrate this type of measurement. At increasing crack lengths, the distributed wavelengths (proportional to axial strain) are measured by an FBG embedded parallel to the delamination plane. The long gauge length of the sensor provides a sufficiently large set of data so that the crack position and growth direction can be distinguished. The strains retrieved from these experiments are further employed to determine the stress distribution caused by the fibres bridging the delamination crack. The combination of FBG measurements with inverse identification via finite element modelling is a new technique for determining bridging laws from static delamination specimens. Results of this work indicate that the maximum bridging stress is approximately 2.5 MPa and that the fibre bridging zone length ranges from 20-50 mm. Comparisons of bridging laws determined using this method and a J-integral approach are made using a second finite element model that includes cohesive elements. Simulations of advancing delamination cracks highlight the sensitivity of the force-displacement response of the specimen to differences in bridging laws. Through the advances in FBG-based methods outlined in this thesis, significant progress is made in the area of non-homogeneous strain detection in fibre-reinforced composites. This allows for improved characterization of three-dimensional residual strain states and the non-uniform strain distributions caused by delamination cracking.

EPFL2007

Reduced basis method for linear elasticity problems with many parameters

Alfio Quarteroni, Gianluigi Rozza

he reduced basis (RB) methods are proposed here for the solution of parametrized equations in linear elasticity problems. The fundamental idea underlying RB methods is to decouple the generation and projection stages (offline/online computational procedures) of the approximation process in order to solve parametrized equations in a rapid, inexpensive and reliable way. The method allows important computational savings with respect to the classical Galerkin-finite element method, ill suited to a repetitive environment like the parametrized contexts of optimization, many queries and sensitivity analysis. We consider different parametrization for the systems: either physical quantities --to model the materials and loads-- and geometrical parameters --to model different geometrical configurations--. Then we describe three different applications of the method in problems with isotropic and orthotropic materials working in plane stress and plane strain approximation and subject to harmonic loads.