Compliance and Hill polarization tensor of a crack in an anisotropic matrix

Barthélémy, Jean‐François

doi:10.1016/j.ijsolstr.2009.08.003

Cited by 17 publications

(26 citation statements)

References 41 publications

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“…Let us consider the two global unit orthogonal vectors e 1 and e 2 in the plane ξ 3 = 0; any unit vector in this plane reads then ξ = cos ψe 1 + sin ψe 2 . It follows that the acoustic tensor K = ξ.C.ξ takes the form:…”

Section: A Proposed Methodology For the Determination Of P Tensormentioning

confidence: 99%

“…We begin with a brief description of the Mori-Tanaka homogenization scheme, that will be further used to obtain the effective compliance (2). The focus of section 3 is the derivation of an analytic expression of the Eshelby tensor for 3D microcracks systems.…”

Section: Introductionmentioning

confidence: 99%

“…It is convenient to note that the developed approach has been recently followed and extended on some aspects by [33] and by [2] who proposed semi analytical techniques for determining the effective compliance for higher anisotropic matrix containing inclusions.…”

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confidence: 99%

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A micromechanical approach of crack-induced damage in orthotropic media: Application to a brittle matrix composite

Monchiet¹,

Gruescu²,

Cazacu³

et al. 2012

Engineering Fracture Mechanics

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. A micromechanical approach of crackinduced damage in orthotropic media : application to a brittle matrix composite. Engineering Fracture Mechanics, Elsevier, 2012, 83 (-) AbstractCoupling between initial and damage-induced anisotropies in 3D elastic damaged materials has been so far addressed by homogenization techniques only for particular microcracks configurations. The main difficulty in developing a general 3D micromechanical model lies in the lack of a closed-form solution of the Eshelby tensor corresponding to cracks in an elastic anisotropic medium. In this study, we begin to present closed form expressions of the Eshelby tensor S that we derived for a cylindrical crack embedded in an orthotropic material. The 3D obtained expressions reduce to existing results in the 2D case or in particular 3D cracks configurations. The effective compliance of an orthotropic medium containing arbitrarily oriented cracks is then derived by using the new Eshelby tensor. Finally, a damage model is formulated by combining the above results with classical thermodynamics approach.The ability of the model to capture coupling between initial orthotropy and damage induced anisotropy is demonstrated through a comparison with experimental data available for a ceramic matrix composite (unidirectional SiC-SiC).

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Section: A Proposed Methodology For the Determination Of P Tensormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A micromechanical approach of crack-induced damage in orthotropic media: Application to a brittle matrix composite

Monchiet¹,

Gruescu²,

Cazacu³

et al. 2012

Engineering Fracture Mechanics

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“…where the superscript 𝑑 stands for disk, since the limit 𝑋 → 0 has been taken. This tensor is computed from the 𝑂 (𝑋) term in the expansion of Q 𝐼,𝑚 , as shown by the theorem proved in Barthelemy [3] (see appendix of that reference). The important result of the theorem is that U 𝑑,0 is also singular, and its kernel is…”

Section: Flat Voidsmentioning

confidence: 99%

Contribution of FE and FFT-based methods to the determination of the effective elastic and conduction properties of composite media with flat inclusions and infinite contrast

Bluthé

Bary

Lemarchand

2021

International Journal of Solids and Structures

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“…The elastic behavior of a medium with randomly oriented cracks has been exhaustively studied in the literature (Barthélémy, 2009;Budiansky and O'Connell, 1976;Henyey and Pomphrey, 1982;Hudson, 1980;Kachanov, 1993;Kachanov et al, 1994;Kemeny and Cook, 1986;Sayers and Kachanov, 1991;Schoenberg and Sayers, 1995;Walsh, 1965). However, natural fractures are not randomly distributed within the Soultz EGS site (Valley, 2007).…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Crack Features and Shear-Wave Splitting Associated with Fracture Extension during Hydraulic Stimulation of the Geothermal Reservoir in Soultz-sous-Forêts

Adelinet

Dorbath

Calò

et al. 2015

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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-The recent tomography results obtained within the scope of the Enhanced Geothermal System (EGS) European Soultz project led us to revisit the meso-fracturing properties of Soultz test site. In this paper, we develop a novel approach coupling effective medium modeling and shear-wave splitting to characterize the evolution of crack properties throughout the hydraulic stimulation process. The stimulation experiment performed in 2000 consisted of 3 successive injection steps spanning over 6 days. An accurate 4-D tomographic image was first carried out based upon the travel-times measured for the induced seismicity [Calò M., Dorbath C., Cornet F.H., Cuenot N. (2011) Large-scale aseismic motion identified through 4-D P-wave tomography, Geophys. J. Int. 186, 1295Int. 186, -1314. The current study shows how to take advantage of the resulting compressional wave (Calò et al., 2011) and shear-wave velocity models. These are given as input data to an anisotropic effective medium model and converted into crack properties. In short, the effective medium model aims to estimate the impact of cracks on velocities. It refers to a crack-free matrix and 2 families of penny-shaped cracks with orientations in agreement with the main observed geological features: North-South strike and dip of 65°East and 65°West [Genter A., Traineau H. (1996) Analysis of macroscopic fractures in granite in the HDR geothermal well EPS-1, Soultz-sousForêts, France, J. Vol. Geoth. Res. 72, 121-141], respectively. The resulting output data are the spatial distributions of crack features (lengths and apertures) within the 3-D geological formation. We point out that a flow rate increase results in a crack shortening in the area imaged by both compressional and shear waves, especially in the upper part of the reservoir. Conversely, the crack length, estimated during continuous injection rate phases, is higher than during the increasing injection rate phases. A possible explanation for this is that cracks remain large because the system has time to relax. We also calculate the extension and opening rates during all hydraulic stimulation sets. While the opening rate is unchanged, the extension rate varies depending on the stimulation phase. It is also shown to be higher around and above the open-hole section than below. This can indicate a potential upward path that makes fluid percolation easier within the granite formation, this path being induced by the temperature gradient. We also compare the evolution of crack extension during injection with shear-wave splitting. Split shear waves were recorded at 2 stations during hydraulic stimulation and processed in terms of splitting parameters. The fast shear-wave polarization remains constant and parallel to the maximum horizontal stress orientation while the amplitude of splitting varies with time. We observe a good agreement between travel-time differences and crack extension rates during the first 4 days of the stimulation experiment. Afterwards, these two parameters depart from each other. This stud...

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Compliance and Hill polarization tensor of a crack in an anisotropic matrix

Cited by 17 publications

References 41 publications

A micromechanical approach of crack-induced damage in orthotropic media: Application to a brittle matrix composite

A micromechanical approach of crack-induced damage in orthotropic media: Application to a brittle matrix composite

Contribution of FE and FFT-based methods to the determination of the effective elastic and conduction properties of composite media with flat inclusions and infinite contrast

Crack Features and Shear-Wave Splitting Associated with Fracture Extension during Hydraulic Stimulation of the Geothermal Reservoir in Soultz-sous-Forêts

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