A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

With the aid of conformal mapping and analytic continuation, we prove that within the framework of anti-plane elasticity, a non-parabolic open elastic inhomogeneity can still admit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding elastic matrix is subjected to uniform remote stresses. The non-circular inclusion can take the form of a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Our analysis indicates that the uniform stress field within the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion. On the other hand, the non-parabolic shape of the inhomogeneity is caused solely by the presence of the non-circular inclusion.

Calculations on compact disc cracking

The Griffith equation for brittle cracking has three problems. First, it applies to an infinite sheet whereas a laboratory test sample is typically near 100 × 100 mm. Second, it describes a central crack instead of the more dangerous and easily observable edge crack. Third, the theory assumes a uniform stress field, instead of tensile force application used in the laboratory. The purpose of this paper is to avoid these difficulties by employing Gregory's solution in calculating the crack behaviour of PMMA (Poly Methyl Meth Acrylate) discs, pin loaded in tension. Our calculations showed that axial disc loading gave nominal strengths comparable with Griffith theory, but the force went to zero as the crack fully crossed the disc, fitting experimental results. Off-axis loading was more interesting because the predicted strength was lower than in axial testing, but also gave unexpected behaviour at short crack lengths, where nominal strength did not rise indefinitely but dropped as crack length went below D/10, quite different from Griffith, where strength rose continuously as cracks were shortened. Such off-axis loading leads to a size effect in which larger discs are weaker, reminiscent of the fine fibre strengthening phenomenon reported in Griffith's early paper (Griffith 1921 Phil. Trans. R. Soc. Lond. A 221 , 163–198. ( doi:10.1098/rsta.1921.0006 )). This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction'.

A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

We use conformal mapping techniques together with analytic continuation to show that a non-parabolic open elastic inhomogeneity continues to admit a state of uniform internal stress when a hole with closed curvilinear traction-free boundary is placed in its vicinity and the surrounding matrix is subjected to uniform remote anti-plane stresses. The internal uniform stress field inside the inhomogeneity is found to be independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In contrast, the non-parabolic shape of the inhomogeneity is influenced solely by the existence of the nearby hole.

Seismic imaging of St Gallen (Switzerland) deep geothermal field medium properties

<p>Sub-surface operations for energy production may originate various environmental risks among which, of great relevance is the seismic risk due to the induced seismicity associated with field operations.</p><p>In the framework of the H2020 Science4CleanEnergy project, S4CE, a multi-disciplinary project aimed at understanding the underlying physical mechanisms underpinning sub-surface geo-energy operations and to measure, control and mitigate their environmental risks, we have investigated the role of fluids in the generation of the seismicity induced during the deep geothermal drilling project close to the city of St.Gallen, Switzerland. To this aim we applied the Focal Mechanism Tomography (FMT) technique and the velocity and attenuation tomography using data collected by the Swiss Seismological Service in 2013 while realizing well control measures after drilling and acidizing the GT-1 well. The dataset consists of 347 earthquakes with magnitude (M<sub>L</sub><sup>corr</sup>) between -1.2 and 3.5. P and S phases were initially hand-picked on three-component ground velocity recordings. As an additional enhancement, a refined re-picking algorithm based on the waveforms cross-correlation was applied providing accurate travel-times data set. The revised picks and P polarities were used both to re-locate the events, using probabilistic approach considering both the absolute both the differential arrival times, and to estimate fault mechanisms using the FPFIT code. Only those events having at least 6 clear P-wave polarities have been analysed. To better constrain the focal mechanisms, for the larger magnitude events the BISTROP code (Bayesian Inversion of Spectral-Level Ratios and P-Wave Polarities) has been also applied.</p><p>Using the FMT technique we estimated the 3D excess pore fluid pressure field at the events hypocentre. Basically, the technique assumes that fault strength is controlled by Coulomb failure criterion and, under the hypothesis of uniform stress field, it ascribes the focal mechanism variations to pore fluid pressure acting on faults.</p><p>The velocity model and the attenuation model have been estimated by using an iterative tomographic inversion of P and S arrival times and t* quantities, which are defined as the ratio of the travel time and quality factor (Q). The t* measures for both P and S wave have been obtained from the analysis of the displacement spectra. We found that fault mechanisms do not fit a uniform stress-field. Based on the events depth, at least two different stress-fields are required. FMT results indicate that fluids contributed to the generation of the induced events. Taking into account for the uncertainties, the inferred excess pore fluid pressure is consistent with the wellhead pressure. Moreover, a correlation exists between the high excess pore fluid pressure and the high Vp/Vs values.</p><p> </p><p>This work has been supported by S4CE ("Science for Clean Energy") project, funded from the European Union’s Horizon 2020 - R&I Framework Programme, under grant agreement No 764810 and by MATISSE project funded by Italian Ministry of Education and Research.</p>

Elastic-Plastic-Damaged Zones around a Deep Circular Wellbore under Non-Uniform Loading

Wellbores are largely constructed during coal mining, shale gas production, and geothermal exploration. Studying the shape and size of the disturbed zone in surrounding rock is of great significance for wellbore stability control. In this paper, a theoretical model for elastic-plastic-damage analysis around a deep circular wellbore under non-uniform compression is proposed. Based on the elastoplastic softening constitutive model and Mohr-Coulomb strength criterion, the analytical expressions of stresses in the elastic, plastic and damaged zones around a circle wellbore are derived. Further, the boundary line equations among the three zones are obtained according to the conditions of stress continuity. Then, the influence rules of non-uniform in-situ stress and mechanical parameters on the stress distribution and plastic zone size in surrounding rock mass are analyzed. The plastic and the damaged zones are both approximately elliptical in shape. When the lateral stress coefficient of the in-situ stress field takes the value 1, the model degenerates into the Yuan Wenbo’s Solution. If the brittleness coefficient of the surrounding rock is 0, the model degenerates into the Kastner’s Equation. Finally, the results are compared with those under two special cases (in the elastoplastic softening rock under a uniform stress field, in the ideal elastoplastic rock under a non-uniform stress field) and a common approximation method (the perturbation method).

A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

Using conformal mapping techniques and analytic continuation, we prove that when subjected to anti-plane elastic deformations, a non-parabolic open inhomogeneity continues to admit an internal uniform stress field when a circular Eshelby inclusion is placed in its vicinity and the surrounding matrix is subjected to uniform remote stresses. Explicit expressions for the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion are obtained. The internal uniform stress field is independent of the shape of the inhomogeneity and the presence of the circular Eshelby inclusion, whereas the existence of the circular Eshelby inclusion exerts a significant influence on the shape of the non-parabolic open inhomogeneity as well as on the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion itself.

A generalisation to cohesive cracks evolution under effects of non-uniform stress field

The aim of the present work is to study the stabilizing effect of the non-uniformity of the stress field on the cohesive cracks evolution in two-dimensional elastic structures. The crack evolution is governed by Dugdale's or Barenblatt's cohesive force models. We distinguish two stages in the crack evolution: the first one where all the crack is submitted to cohesive forces, followed by a second one where a non cohesive part appears. Assuming that the material characteristic length dc associated with the cohesive model is small by comparison to the dimension L of the body, we develop a two-scale approach, and using the complex analysis method, we obtain the entire crack evolution with the loading in a closed form for the Dugdale's case and in semi-analytical form for the Barenblatt's case. In particular, we show that the propagation is stable during the first stage, but becomes unstable with a brutal jump of the crack length as soon as the non cohesive crack part appears. We discuss also the influence of all the parameters of the problem, in particular the non-uniform stress and cohesive model formulations, and study the sensitivity to imperfections.