In order to evaluate the performance of deep geological disposal of radioactive waste,an underground research laboratory(URL)was constructed by Andra in the Callovo-Oxfordian(COx)claystone formation at the Meuse/Haute...In order to evaluate the performance of deep geological disposal of radioactive waste,an underground research laboratory(URL)was constructed by Andra in the Callovo-Oxfordian(COx)claystone formation at the Meuse/Haute-Marne(MHM).The construction of URL induced the excavation damage of host formations,and the ventilation in the galleries desaturated the host formation close to the gallery wall.Moreover,it is expected that the mechanical behaviour of COx claystone is time-dependent.This study presents a constitutive model developed to describe the viscoplastic behaviour of unsaturated and damaged COx claystone.In this model,the unsaturation effect is considered by adopting the Bishop effective stress and the van Genuchten(VG)water retention model.In terms of the viscoplastic behaviour,the nonstationary flow surface(NSFS)theory for unsaturated soils is used with consideration of the coupled effects of strain rate and suction on the yield stress.A progressive hardening law is adopted.Meanwhile,a non-associated flow rule is used,which is similar to that in Barcelona basic model(BBM).In addition,to describe the damage effect induced by suction change and viscoplastic loading,a damage function is defined based on the crack volume proportion.This damage function contains two variables:unsaturated effective stress and viscoplastic volumetric strain,with the related parameters determined based on the mercury intrusion porosimetry(MIP)tests.For the model validation,different tests on COx claystone under different loading paths are simulated.Comparisons between experimental and simulated results indicated that the present model is able to well describe the viscoplastic behaviour of damaged COx claystone,including swelling/shrinkage,triaxial extension and compression,and triaxial creep.展开更多
The moisture content of subgrade soil in seasonally frozen regions is often higher than its optimum value,leading to a decline in mechanical properties and a reduction in subgrade bearing capacity.Electro-osmosis has ...The moisture content of subgrade soil in seasonally frozen regions is often higher than its optimum value,leading to a decline in mechanical properties and a reduction in subgrade bearing capacity.Electro-osmosis has shown promise as a technology for controlling subgrade moisture,but significant heterogeneity has also been observed in treated soil.This study investigates the impact of electro-osmosis on soil stiffness through a series of bender element tests of compacted clay.The effects of dry density and supply voltage on the performance of electroosmosis treatment and the layered structure and anisotropy of the soil were analyzed.The results show that electro-osmosis treatment increased the shear wave velocity of the soil by 140% compared to untreated saturated soil and by 70% compared to soil with optimum water content.It has also been found that layered compaction of soil resulted in a layered structure,with electro-osmosis having a more prominent impact on soil near the cathode,resulting in a more pronounced layered structure.Besides,electro-osmosis was found to enhance soil anisotropy,particularly near the anode.Increasing the dry density and voltage levels can help improve soil uniformity.These findings provide insights into the potential use of electro-osmosis in improving soil stiffness,which could benefit various engineering applications.展开更多
When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) &ra...When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.展开更多
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonli...In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.展开更多
Bentonite-based materials have been considered in many countries as engineered barrier/backfilling materials in deep geological disposal of high-level radioactive waste.During the long period of waste storage,these ma...Bentonite-based materials have been considered in many countries as engineered barrier/backfilling materials in deep geological disposal of high-level radioactive waste.During the long period of waste storage,these materials will play an essential role in ensuring the integrity of the storage system that consists of the waste canisters,the engineered barrier/backfill,the retaining structures as well as the geological barrier.Thus,it is essential to well understand the hydro-mechanical behaviours of these bentonite-based materials.This review paper presents the recent advances of knowledge on MX80 bentonite-based materials,in terms of water retention properties,hydraulic behaviour and mechanical behaviour.Emphasis is put on the effect of technological voids and the role of the dry density of bentonite.The swelling anisotropy is also discussed based on the results from swelling tests with measurements of both axial and radial swelling pressures on a sand-bentonite mixture compacted at different densities.Microstructure observation was used to help the interpretation of macroscopic hydromechanical behaviour.Also,the evolution of soil microstructure thus the soil density over time is discussed based on the results from mock-up tests.This evolution is essential for understanding the longterm hydro-mechanical behaviour of the engineered barrier/backfill.展开更多
An overview of the recent findings about the chemo-hydro-mechanical behaviour of materials used for both geological and engineered barriers in nuclear waste disposal is presented, through some examples about the natur...An overview of the recent findings about the chemo-hydro-mechanical behaviour of materials used for both geological and engineered barriers in nuclear waste disposal is presented, through some examples about the natural Boom Clay （BC） and compacted bentonite-based materials. For the natural BC, it was found that compression index identified from both oedometer and isotropic compression tests is sim- ilar and the compressibility of BC from the Mol site is higher than that of BC from the Essen site： the shear strength of Mol BC is also higher than that of the Essen BC, suggesting a significant effect of carbo- nates content; the thermal volume change is strongly overconsolidation ratio （OCR） dependent-low OCR values promote thermal contraction while high OCR values favour thermal dilation; the volume change behaviour is also strongly time dependent and this time dependent behaviour is governed by the stress level and temperature; the effect of pore-water salinity on the volume change behaviour can be signif- icant when the smectite content is relatively high. For the bentonite-based materials, it was found that thermal contraction also occurs at low OCR values, but this is suction dependent--suction promotes ther- mal dilation. Under constant volume conditions, wetting results in a decrease of hydraulic conductivity, followed by an increase. This is found to be related to changes in macro-pores size-wetting induces a decrease of macro-pores size, followed by an increase due to the aggregates fissuring. The presence of technological voids can increase the hydraulic conductivity but does not influence the swelling pressure.展开更多
Clays and argillites are considered in some countries as possible host rocks for nuclear waste disposal at great depth.The use of compacted swelling clays as engineered barriers is also considered within the framework...Clays and argillites are considered in some countries as possible host rocks for nuclear waste disposal at great depth.The use of compacted swelling clays as engineered barriers is also considered within the framework of the multi-barrier concept.In relation to these concepts,various research programs have been conducted to assess the thermo-hydro-mechanical properties of radioactive waste disposal at great depth.After introducing the concepts of waste isolation developed in Belgium,France and Switzerland,the paper describes the retention and transfer properties of engineered barriers made up of compacted swelling clays in relation to microstructure features.Some features of the thermo-mechanical behaviors of three possible geological barriers,namely Boom clay（Belgium）,Callovo-Oxfordian clay（France） and Opalinus clay（Switzerland）,are then described,including the retention and transfer properties,volume change behavior,shear strength and thermal aspects.展开更多
It is well-known that a close link exists between soil-water retention curve(SWRC)and pore size distribution(PSD).Theoretically,mercury intrusion porosimetry(MIP)test simulates a soil drying path and the test results ...It is well-known that a close link exists between soil-water retention curve(SWRC)and pore size distribution(PSD).Theoretically,mercury intrusion porosimetry(MIP)test simulates a soil drying path and the test results can be used to deduce the SWRC(termed SWRCMIP).However,SWRCMIP does not include the effect of volume change,compared with the conventional SWRC that is directly determined by suction measurement or suction control techniques.For deformable soils,there is a significant difference between conventional SWRC and SWRCMIP.In this study,drying test was carried out on a reconstituted silty soil,and the volume change,suction,and PSD were measured on samples with different water contents.The change in the deduced SWRCMIP and its relationship with the conventional SWRC were analyzed.The results showed that the volume change of soil is the main reason accounting for the difference between conventional SWRC and SWRCMIP.Based on the test results,a transformation model was then proposed for conventional SWRC and SWRCMIP,for which the soil state with no volume change is taken as a reference.Comparison between the experimental and predicted SWRCs showed that the proposed model can well consider the influence of soil volume change on its water retention property.展开更多
The soil-atmosphere interaction was investigated through laboratory testing,field monitoring and numerical monitoring.In the laboratory,the soil water evaporation mechanisms were studied using an environmental chamber...The soil-atmosphere interaction was investigated through laboratory testing,field monitoring and numerical monitoring.In the laboratory,the soil water evaporation mechanisms were studied using an environmental chamber equipped with a large number of sensors for controlling both the air parameters and soil parameters.Both sand and clay were considered.In case of sand,a dry layer could be formed during evaporation in the near surface zone where the suction corresponded to the residual volumetric water content.The evaporative surface was situated at a depth where the soil temperature was the lowest.In case of clay,soil cracking occurred,changing the evaporative surface from one-dimensional to three-dimensional nature.The suctionbased evaporation model was adapted to take these phenomena into account by adopting a function of dry layer evolution in the case of sand and by adopting a surface crack ratio and a retative humidity ratio in the case of clay.In the field,the volumetric water content,and the suction as well as the runoff were monitored for an embankment constructed with lime/cement treated soils.It appeared that using precipitation data only did not allow a correct description of the variations of volumetric water content and suction inside the soils,the consideration of water evaporation being essential.It was possible to use a correlation between precipiration and runoff.The hydraulic conductivity was found to be a key parameter controlling the variations of volumetric water content and suction.For the numerical modelling,a fully coupled thermohydraulic model was developed,allowing analyzing the changes in temperature,volumetric water content and suction of soil,with the upper boundary conditions at the interface between soil and atmosphere determined using meteorological data.Comparison between simulations and measurements showed the performance of such numerical approach.展开更多
To better understand the salinity effect on the compaction behaviour of soil,standard Proctor compaction test was conducted on soil samples with different salinities.Matric suction and small-strain shear modulus,G_(ma...To better understand the salinity effect on the compaction behaviour of soil,standard Proctor compaction test was conducted on soil samples with different salinities.Matric suction and small-strain shear modulus,G_(max),were determined and pore size distribution was also investigated on samples statically compacted at different water contents.Results showed that with the decrease of soil salinity from initial value of 2.1‰(g of salt/kg of dry soil)to zero,the maximum dry density increased and the optimum water content decreased,whereas there was no significant change with the increase of soil salinity from 2.1‰ to 6.76‰.Interestingly,it was observed that G_(max) also decreased when the soil salinity decreased from initial value of 2.1‰ to zero and kept almost constant when the soil salinity increased from 2.1‰ to 6.76‰,for dry samples with similar matric suction and also for samples compacted at optimum state and on wet side whose matric suctions were slightly different due to the difference in remoulded water content.Furthermore,the effect of salinity on compaction behaviour and G_(max) decreased for samples compacted from dry side to wet side.The pore size distribution exhibited bi-modal characteristics with two populations of micro-and macro-pores not only for samples compacted on dry side and at optimum state,but also for those compacted on wet side.Further examination showed that the modal size of micro-pores shifted to lower values and that of macro-pores shifted to higher values for saline soil compared to the soil without salt.展开更多
Highly compacted expansive clays have been usually considered as a possible material for sealing and backfill in deep geological disposal of radioactive waste. In this condition, the material is simultaneously subject...Highly compacted expansive clays have been usually considered as a possible material for sealing and backfill in deep geological disposal of radioactive waste. In this condition, the material is simultaneously subjected to water infiltration from the geological barrier and stresses generated by the swelling of engineered barriers in confined conditions. Its behaviour under hydro-mechanical loading is essential to the safe design of the whole storage system. In the present work, MX80 bentonite, a kind of expansive clay from Wyoming, USA, was studied. After compaction, its dry density was 1.8 Mg/m^3 and its initial suction was 110 MPa. Firstly, the soil was humidified under controlled suction and free-swelling conditions. Significant swelling was observed. Secondly, four values of suction of 110, 39, 9 and 0 MPa were employed to perform isotropic compressive tests at constant suction conditions. That allowed studying the effect of suction on the yield pressure, elastic and plastic compressibility parameters. The results show that the elastic and plastic compressibility parameters increase when the suction decreases. The relationship between these parameters and the logarithm of suction can be linearly correlated. The yield stress drastically decreases upon wetting under free-swelling conditions, from 12 - 18 MPa （at an initial suction of 110 MPa） to 0.2 MPa at saturated state.展开更多
Thermal conductivity, water retention curve and swelling behavior of two MX80 bentonite samples were studied in the present work. The difference obtained from these two MX80 bentonite samples was then analyzed in term...Thermal conductivity, water retention curve and swelling behavior of two MX80 bentonite samples were studied in the present work. The difference obtained from these two MX80 bentonite samples was then analyzed in terms of mineralogical effects （effects of the proportion of quartz and montmorillonite）. It was concluded that the mineralogical effect was significant on the thermal conductivity and the swelling capacity; on the contrary, it was negligible on the water retention property.展开更多
Pre-compacted elements （disks, tutus） of bentonite/sand mixture are candidate materials for sealing plugs of radioactive waste disposal. Choice of this material is mainly based on its swelling capacity allowing all ...Pre-compacted elements （disks, tutus） of bentonite/sand mixture are candidate materials for sealing plugs of radioactive waste disposal. Choice of this material is mainly based on its swelling capacity allowing all gaps in the system to be sealed, and on its low permeability. When emplaced in the gallery, these elements will start to absorb water from the host rock and swell. Thereby, a swelling pressure will develop in the radial direction against the host rock and in the axial direction against the support structure. In this work, the swelling pressure of a small scale compacted disk of bentonite and sand was experimentally studied in both radial and axial directions. Different swelling kinetics were identified for different dry densities and along different directions. As a rule, the swelling pressure starts increasing quickly, reaches a peak value, decreases a little and finally stabilises. For some dry densities, higher peaks were observed in the radial direction than in the axial direction. The presence of peaks is related to the microstructure change and to the collapse of macro- pores. In parallel to the mechanical tests, microstructure investigation at the sample scale was conducted using microfocus X-ray computed tomography （BCT）. Image observation showed a denser structure in the centre and a looser one in the border, which was also confirmed by image analysis. This structure heterogeneity in the radial direction and the occurrence of macro-pores close to the radial boundary of the sample can explain the large peaks observed in the radial swelling pressure evolution. Another interesting result is the higher anisotropy found at lower bentonite dry densities, which was also analysed by means ofμCT observation of a sample at low bentonite dry density after the end of test. It was found that the macro-pores, especially those between sand grains, were not filled by swelled bentonite, which preserved the anisotropic microstructure caused by uniaxial compression due to the absence of microstructure collapse.展开更多
In the French deep geological disposal for radioactive wastes,compacted bentonite/claystone mixtures have been considered as possible sealing materials.After emplacement in place,such mixtures are hydrated by the site...In the French deep geological disposal for radioactive wastes,compacted bentonite/claystone mixtures have been considered as possible sealing materials.After emplacement in place,such mixtures are hydrated by the site solution as well as the cement solution produced by the degradation of concrete.In this study,the effects of synthetic site solution and cement solution on the hydro-mechanical behaviour of compacted mixtures of claystone and two types of bentonites(MX80 Na-bentonite and Sardinia Cabentonite)were investigated by carrying out a series of swelling pressure,hydraulic conductivity and mercury intrusion porosimetry(MIP)tests.It was found that for the MX80 bentonite/claystone mixture hydrated with synthetic site solution,the swelling capacity was reduced compared to the case with deionised water owing to the transformation of Na-montmorillonite to multi-cation dominant montmorillonite by cation exchanges.For the Sardinia bentonite/claystone mixture,the similar increasing rate of swelling pressure was observed during the crystalline swelling process for different solutions,suggesting insignificant cation exchanges.Additionally,the cations in the synthetic site solution could reduce the thickness of diffuse double layer and the osmotic swelling for both MX80 bentonite/claystone and Sardinia bentonite/claystone mixtures.The large-pore volume increased consequently and enhanced water flow.In the cement solution,the hydroxide could also dissolve the montmorillonite,reducing the swelling pressure,and increase the large-pore volume,facilitating the water flow.Furthermore,the decrease of swelling pressure and the increase of hydraulic conductivity were more significant in the case of low dry density because of more intensive interaction between montmorillonite and hydroxide due to the high permeability.展开更多
This study aims at evidencing the effects of lime treatment on the microstructure and hydraulic conductivityof a compacted expansive clay, with emphasis put on the effect of lime hydration and modification.For this pu...This study aims at evidencing the effects of lime treatment on the microstructure and hydraulic conductivityof a compacted expansive clay, with emphasis put on the effect of lime hydration and modification.For this purpose, evolutions of hydraulic conductivity were investigated for both lime-treatedand untreated soil specimens over 7 d after full saturation of the specimens and their microstructureswere observed at the end. Note that for the treated specimen, dry clay powder was mixed with quicklimeprior to compaction in order to study the effect of lime hydration. It is observed that lime hydration andmodification did not affect the intra-aggregate pores but increased the inter-aggregates pores size. Thisincrease gave rise to an increase of hydraulic conductivity. More precisely, the hydraulic conductivity oflime-treated specimen increased progressively during the first 3 d of modification phase and stabilisedduring the next 4 d which correspond to a short period prior to the stabilisation phase. The microstructureobservation showed that stabilisation reactions took place after 7 d. Under the effect of stabilisation,a decreasing hydraulic conductivity can be expected in longer time due to the formation ofcementitious compounds. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.展开更多
During the constructions of motorways and high-speed railway lines in the Yanji Basin,large amounts of excess mudstones due to the enormous tunnel excavations and slope cuts would be deposited as landfills.Assessing t...During the constructions of motorways and high-speed railway lines in the Yanji Basin,large amounts of excess mudstones due to the enormous tunnel excavations and slope cuts would be deposited as landfills.Assessing the deformation and permeability of Yanji mudstone became important for the design,construction and operation of the landfills.This paper presents an experimental study on the deformation and permeability of Yanji mudstone by carrying out a series of oedometer tests with loading/unloading cycles.The results show that the sample with a lower initial water content exhibited greater swelling deformation after inundation,a lower yield stress,greater deformation and a higher hydraulic conductivity during the loading/unloading cycles.As the number of loading/unloading cycles increased,the yield stress and accumulated plastic deformation increased,while the compression index,rebound index and hydraulic conductivity decreased.The samples became stiffer and their hydromechanical behaviour tended to be stable after three cycles.The compression curves could be divided into pre-yield and post-yield zones.The post-yield zones of compression curves and the rebound curves could be normalized into a unique line,and the pre-yield zones of the compression curves could be described as lines.Basic equations were developed to predict mudstone deformation under cyclic loading and unloading.Additionally,an empirical relationship between the hydraulic conductivity and void ratio was also proposed.The ability of the proposed methods was verified by the overall good agreement between the experimental results and predicted values.展开更多
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in ...The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.展开更多
A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well know...A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper (DOI:10.4236/jmp.2018.910125) we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S(<em>m</em>) and K(<em>m</em>, <em>a</em>) are depending on constant parameters in such a way that S <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0 and K<span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><span style="white-space:nowrap;"><span style="white-space:nowrap;"></span></span> S when <em>a</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0, the CC of S do not provide the CC of M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0 while the CC of K do not provide the CC of S when a <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the <em>prolongation/projection</em> (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebra and that the role played by the Spencer operator is crucial. We get K < S < M with 2 < 4 < 10 for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even if each Spencer sequence is isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by the corresponding Lie algebra.展开更多
When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em&...When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-space:nowrap;"><span style="white-space:nowrap;">ä</span></span>cklund problem</em> during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit <em>differential homological algebra</em> by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when <em>M</em> is a <em>D</em>-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;"><em>R</em> = <em>hom<sub>K</sub></em> (<em>M</em>,<em>K</em>)</span> is a <em>D</em>-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.展开更多
When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em...When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators with coefficients in a differential field <em>K</em>, then <em>D</em> gives rise by residue to a <em>differential module M</em> over<em> D</em> while <em>a</em><em style="white-space:normal;">d </em><span style="white-space:normal;">(</span><em style="white-space:normal;">D</em><span style="white-space:normal;">)</span> gives rise to a differential module <em>N =ad (M)</em> over <em>D</em>. The <em>differential extension modules</em> <img src="Edit_55629608-629e-4b52-ac8f-52470473af77.png" alt="" /> with <span style="white-space:nowrap;"><em>ext<span style="font-size:10px;"><sup>0</sup></span></em><em>(M) = hom</em><sub><em>D</em></sub><em> (M, D)</em></span> only depend on <em>M</em> and are measuring the above gaps, <em>independently of the previous differential sequence</em>, in such a way that <span style="white-space:nowrap;"><em>ext</em><sup><em>1</em></sup><em> (N) = t (M)</em> </span> is the torsion submodule of <em>M</em>. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension <em>n</em> and to prove for the first time that the extension modules highly depend on the Vessiot <em>structure constants c</em>. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint <span style="white-space:nowrap;"><em>Beltrami = ad (Riemann)</em></span> which can indeed be parametrized by the operator <span style="white-space:nowrap;"><em>Lanczos = ad (Bianchi) </em></span>in arbitrary dimension, “<em>one step further on to the right</em>” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when <em>D</em> is a Lie operator of finite type, we shall prove that <img src="Edit_3a20593a-fffe-4a20-a041-2c6bb9738d5d.png" alt="" />. It will follow that the <em>Riemann-Lanczos </em>and <em>Weyl-Lanczos</em> problems just amount to prove such a result for <em>i </em>= 1,2 and arbitrary <em>n</em> when <em>D</em> is the <em>classical or conformal Killing</em> operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.展开更多
文摘In order to evaluate the performance of deep geological disposal of radioactive waste,an underground research laboratory(URL)was constructed by Andra in the Callovo-Oxfordian(COx)claystone formation at the Meuse/Haute-Marne(MHM).The construction of URL induced the excavation damage of host formations,and the ventilation in the galleries desaturated the host formation close to the gallery wall.Moreover,it is expected that the mechanical behaviour of COx claystone is time-dependent.This study presents a constitutive model developed to describe the viscoplastic behaviour of unsaturated and damaged COx claystone.In this model,the unsaturation effect is considered by adopting the Bishop effective stress and the van Genuchten(VG)water retention model.In terms of the viscoplastic behaviour,the nonstationary flow surface(NSFS)theory for unsaturated soils is used with consideration of the coupled effects of strain rate and suction on the yield stress.A progressive hardening law is adopted.Meanwhile,a non-associated flow rule is used,which is similar to that in Barcelona basic model(BBM).In addition,to describe the damage effect induced by suction change and viscoplastic loading,a damage function is defined based on the crack volume proportion.This damage function contains two variables:unsaturated effective stress and viscoplastic volumetric strain,with the related parameters determined based on the mercury intrusion porosimetry(MIP)tests.For the model validation,different tests on COx claystone under different loading paths are simulated.Comparisons between experimental and simulated results indicated that the present model is able to well describe the viscoplastic behaviour of damaged COx claystone,including swelling/shrinkage,triaxial extension and compression,and triaxial creep.
基金supported by the National Natural Science Foundation of China(No.41971076,No.42171128)。
文摘The moisture content of subgrade soil in seasonally frozen regions is often higher than its optimum value,leading to a decline in mechanical properties and a reduction in subgrade bearing capacity.Electro-osmosis has shown promise as a technology for controlling subgrade moisture,but significant heterogeneity has also been observed in treated soil.This study investigates the impact of electro-osmosis on soil stiffness through a series of bender element tests of compacted clay.The effects of dry density and supply voltage on the performance of electroosmosis treatment and the layered structure and anisotropy of the soil were analyzed.The results show that electro-osmosis treatment increased the shear wave velocity of the soil by 140% compared to untreated saturated soil and by 70% compared to soil with optimum water content.It has also been found that layered compaction of soil resulted in a layered structure,with electro-osmosis having a more prominent impact on soil near the cathode,resulting in a more pronounced layered structure.Besides,electro-osmosis was found to enhance soil anisotropy,particularly near the anode.Increasing the dry density and voltage levels can help improve soil uniformity.These findings provide insights into the potential use of electro-osmosis in improving soil stiffness,which could benefit various engineering applications.
文摘When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.
文摘In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.
文摘Bentonite-based materials have been considered in many countries as engineered barrier/backfilling materials in deep geological disposal of high-level radioactive waste.During the long period of waste storage,these materials will play an essential role in ensuring the integrity of the storage system that consists of the waste canisters,the engineered barrier/backfill,the retaining structures as well as the geological barrier.Thus,it is essential to well understand the hydro-mechanical behaviours of these bentonite-based materials.This review paper presents the recent advances of knowledge on MX80 bentonite-based materials,in terms of water retention properties,hydraulic behaviour and mechanical behaviour.Emphasis is put on the effect of technological voids and the role of the dry density of bentonite.The swelling anisotropy is also discussed based on the results from swelling tests with measurements of both axial and radial swelling pressures on a sand-bentonite mixture compacted at different densities.Microstructure observation was used to help the interpretation of macroscopic hydromechanical behaviour.Also,the evolution of soil microstructure thus the soil density over time is discussed based on the results from mock-up tests.This evolution is essential for understanding the longterm hydro-mechanical behaviour of the engineered barrier/backfill.
文摘An overview of the recent findings about the chemo-hydro-mechanical behaviour of materials used for both geological and engineered barriers in nuclear waste disposal is presented, through some examples about the natural Boom Clay （BC） and compacted bentonite-based materials. For the natural BC, it was found that compression index identified from both oedometer and isotropic compression tests is sim- ilar and the compressibility of BC from the Mol site is higher than that of BC from the Essen site： the shear strength of Mol BC is also higher than that of the Essen BC, suggesting a significant effect of carbo- nates content; the thermal volume change is strongly overconsolidation ratio （OCR） dependent-low OCR values promote thermal contraction while high OCR values favour thermal dilation; the volume change behaviour is also strongly time dependent and this time dependent behaviour is governed by the stress level and temperature; the effect of pore-water salinity on the volume change behaviour can be signif- icant when the smectite content is relatively high. For the bentonite-based materials, it was found that thermal contraction also occurs at low OCR values, but this is suction dependent--suction promotes ther- mal dilation. Under constant volume conditions, wetting results in a decrease of hydraulic conductivity, followed by an increase. This is found to be related to changes in macro-pores size-wetting induces a decrease of macro-pores size, followed by an increase due to the aggregates fissuring. The presence of technological voids can increase the hydraulic conductivity but does not influence the swelling pressure.
文摘Clays and argillites are considered in some countries as possible host rocks for nuclear waste disposal at great depth.The use of compacted swelling clays as engineered barriers is also considered within the framework of the multi-barrier concept.In relation to these concepts,various research programs have been conducted to assess the thermo-hydro-mechanical properties of radioactive waste disposal at great depth.After introducing the concepts of waste isolation developed in Belgium,France and Switzerland,the paper describes the retention and transfer properties of engineered barriers made up of compacted swelling clays in relation to microstructure features.Some features of the thermo-mechanical behaviors of three possible geological barriers,namely Boom clay（Belgium）,Callovo-Oxfordian clay（France） and Opalinus clay（Switzerland）,are then described,including the retention and transfer properties,volume change behavior,shear strength and thermal aspects.
基金Shanghai Key Innovative Team of Cultural Heritage Conservation and the financial support from the National Sciences Foundation of China(Grant Nos.41977214 and 41572284)the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences(Grant No.Z013008)。
文摘It is well-known that a close link exists between soil-water retention curve(SWRC)and pore size distribution(PSD).Theoretically,mercury intrusion porosimetry(MIP)test simulates a soil drying path and the test results can be used to deduce the SWRC(termed SWRCMIP).However,SWRCMIP does not include the effect of volume change,compared with the conventional SWRC that is directly determined by suction measurement or suction control techniques.For deformable soils,there is a significant difference between conventional SWRC and SWRCMIP.In this study,drying test was carried out on a reconstituted silty soil,and the volume change,suction,and PSD were measured on samples with different water contents.The change in the deduced SWRCMIP and its relationship with the conventional SWRC were analyzed.The results showed that the volume change of soil is the main reason accounting for the difference between conventional SWRC and SWRCMIP.Based on the test results,a transformation model was then proposed for conventional SWRC and SWRCMIP,for which the soil state with no volume change is taken as a reference.Comparison between the experimental and predicted SWRCs showed that the proposed model can well consider the influence of soil volume change on its water retention property.
基金the French Research Agency(ANR)within the Project Ter DOUEST ANR-07-PCGU-006-10。
文摘The soil-atmosphere interaction was investigated through laboratory testing,field monitoring and numerical monitoring.In the laboratory,the soil water evaporation mechanisms were studied using an environmental chamber equipped with a large number of sensors for controlling both the air parameters and soil parameters.Both sand and clay were considered.In case of sand,a dry layer could be formed during evaporation in the near surface zone where the suction corresponded to the residual volumetric water content.The evaporative surface was situated at a depth where the soil temperature was the lowest.In case of clay,soil cracking occurred,changing the evaporative surface from one-dimensional to three-dimensional nature.The suctionbased evaporation model was adapted to take these phenomena into account by adopting a function of dry layer evolution in the case of sand and by adopting a surface crack ratio and a retative humidity ratio in the case of clay.In the field,the volumetric water content,and the suction as well as the runoff were monitored for an embankment constructed with lime/cement treated soils.It appeared that using precipitation data only did not allow a correct description of the variations of volumetric water content and suction inside the soils,the consideration of water evaporation being essential.It was possible to use a correlation between precipiration and runoff.The hydraulic conductivity was found to be a key parameter controlling the variations of volumetric water content and suction.For the numerical modelling,a fully coupled thermohydraulic model was developed,allowing analyzing the changes in temperature,volumetric water content and suction of soil,with the upper boundary conditions at the interface between soil and atmosphere determined using meteorological data.Comparison between simulations and measurements showed the performance of such numerical approach.
基金the China Scholarship Council(CSC)Ecole des Ponts ParisTech(ENPC)INRAE for their financial support。
文摘To better understand the salinity effect on the compaction behaviour of soil,standard Proctor compaction test was conducted on soil samples with different salinities.Matric suction and small-strain shear modulus,G_(max),were determined and pore size distribution was also investigated on samples statically compacted at different water contents.Results showed that with the decrease of soil salinity from initial value of 2.1‰(g of salt/kg of dry soil)to zero,the maximum dry density increased and the optimum water content decreased,whereas there was no significant change with the increase of soil salinity from 2.1‰ to 6.76‰.Interestingly,it was observed that G_(max) also decreased when the soil salinity decreased from initial value of 2.1‰ to zero and kept almost constant when the soil salinity increased from 2.1‰ to 6.76‰,for dry samples with similar matric suction and also for samples compacted at optimum state and on wet side whose matric suctions were slightly different due to the difference in remoulded water content.Furthermore,the effect of salinity on compaction behaviour and G_(max) decreased for samples compacted from dry side to wet side.The pore size distribution exhibited bi-modal characteristics with two populations of micro-and macro-pores not only for samples compacted on dry side and at optimum state,but also for those compacted on wet side.Further examination showed that the modal size of micro-pores shifted to lower values and that of macro-pores shifted to higher values for saline soil compared to the soil without salt.
文摘Highly compacted expansive clays have been usually considered as a possible material for sealing and backfill in deep geological disposal of radioactive waste. In this condition, the material is simultaneously subjected to water infiltration from the geological barrier and stresses generated by the swelling of engineered barriers in confined conditions. Its behaviour under hydro-mechanical loading is essential to the safe design of the whole storage system. In the present work, MX80 bentonite, a kind of expansive clay from Wyoming, USA, was studied. After compaction, its dry density was 1.8 Mg/m^3 and its initial suction was 110 MPa. Firstly, the soil was humidified under controlled suction and free-swelling conditions. Significant swelling was observed. Secondly, four values of suction of 110, 39, 9 and 0 MPa were employed to perform isotropic compressive tests at constant suction conditions. That allowed studying the effect of suction on the yield pressure, elastic and plastic compressibility parameters. The results show that the elastic and plastic compressibility parameters increase when the suction decreases. The relationship between these parameters and the logarithm of suction can be linearly correlated. The yield stress drastically decreases upon wetting under free-swelling conditions, from 12 - 18 MPa （at an initial suction of 110 MPa） to 0.2 MPa at saturated state.
文摘Thermal conductivity, water retention curve and swelling behavior of two MX80 bentonite samples were studied in the present work. The difference obtained from these two MX80 bentonite samples was then analyzed in terms of mineralogical effects （effects of the proportion of quartz and montmorillonite）. It was concluded that the mineralogical effect was significant on the thermal conductivity and the swelling capacity; on the contrary, it was negligible on the water retention property.
文摘Pre-compacted elements （disks, tutus） of bentonite/sand mixture are candidate materials for sealing plugs of radioactive waste disposal. Choice of this material is mainly based on its swelling capacity allowing all gaps in the system to be sealed, and on its low permeability. When emplaced in the gallery, these elements will start to absorb water from the host rock and swell. Thereby, a swelling pressure will develop in the radial direction against the host rock and in the axial direction against the support structure. In this work, the swelling pressure of a small scale compacted disk of bentonite and sand was experimentally studied in both radial and axial directions. Different swelling kinetics were identified for different dry densities and along different directions. As a rule, the swelling pressure starts increasing quickly, reaches a peak value, decreases a little and finally stabilises. For some dry densities, higher peaks were observed in the radial direction than in the axial direction. The presence of peaks is related to the microstructure change and to the collapse of macro- pores. In parallel to the mechanical tests, microstructure investigation at the sample scale was conducted using microfocus X-ray computed tomography （BCT）. Image observation showed a denser structure in the centre and a looser one in the border, which was also confirmed by image analysis. This structure heterogeneity in the radial direction and the occurrence of macro-pores close to the radial boundary of the sample can explain the large peaks observed in the radial swelling pressure evolution. Another interesting result is the higher anisotropy found at lower bentonite dry densities, which was also analysed by means ofμCT observation of a sample at low bentonite dry density after the end of test. It was found that the macro-pores, especially those between sand grains, were not filled by swelled bentonite, which preserved the anisotropic microstructure caused by uniaxial compression due to the absence of microstructure collapse.
基金supported by Ecole des Ponts Paris Tech and the French National Radioactive Waste Management Agency(Andra)China Scholarship Council for the grant scholarship。
文摘In the French deep geological disposal for radioactive wastes,compacted bentonite/claystone mixtures have been considered as possible sealing materials.After emplacement in place,such mixtures are hydrated by the site solution as well as the cement solution produced by the degradation of concrete.In this study,the effects of synthetic site solution and cement solution on the hydro-mechanical behaviour of compacted mixtures of claystone and two types of bentonites(MX80 Na-bentonite and Sardinia Cabentonite)were investigated by carrying out a series of swelling pressure,hydraulic conductivity and mercury intrusion porosimetry(MIP)tests.It was found that for the MX80 bentonite/claystone mixture hydrated with synthetic site solution,the swelling capacity was reduced compared to the case with deionised water owing to the transformation of Na-montmorillonite to multi-cation dominant montmorillonite by cation exchanges.For the Sardinia bentonite/claystone mixture,the similar increasing rate of swelling pressure was observed during the crystalline swelling process for different solutions,suggesting insignificant cation exchanges.Additionally,the cations in the synthetic site solution could reduce the thickness of diffuse double layer and the osmotic swelling for both MX80 bentonite/claystone and Sardinia bentonite/claystone mixtures.The large-pore volume increased consequently and enhanced water flow.In the cement solution,the hydroxide could also dissolve the montmorillonite,reducing the swelling pressure,and increase the large-pore volume,facilitating the water flow.Furthermore,the decrease of swelling pressure and the increase of hydraulic conductivity were more significant in the case of low dry density because of more intensive interaction between montmorillonite and hydroxide due to the high permeability.
基金the French National Research Agency for funding the present study within the project-TERDOUEST "Sustainable earthworks involving treated soils"
文摘This study aims at evidencing the effects of lime treatment on the microstructure and hydraulic conductivityof a compacted expansive clay, with emphasis put on the effect of lime hydration and modification.For this purpose, evolutions of hydraulic conductivity were investigated for both lime-treatedand untreated soil specimens over 7 d after full saturation of the specimens and their microstructureswere observed at the end. Note that for the treated specimen, dry clay powder was mixed with quicklimeprior to compaction in order to study the effect of lime hydration. It is observed that lime hydration andmodification did not affect the intra-aggregate pores but increased the inter-aggregates pores size. Thisincrease gave rise to an increase of hydraulic conductivity. More precisely, the hydraulic conductivity oflime-treated specimen increased progressively during the first 3 d of modification phase and stabilisedduring the next 4 d which correspond to a short period prior to the stabilisation phase. The microstructureobservation showed that stabilisation reactions took place after 7 d. Under the effect of stabilisation,a decreasing hydraulic conductivity can be expected in longer time due to the formation ofcementitious compounds. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.
基金financially supported by the National Natural Science Foundation of China (Grant No. 41430634)the State Key Laboratory of Geomechanics and Geotechnical Engineering (Grant No. Y11002Q02)
文摘During the constructions of motorways and high-speed railway lines in the Yanji Basin,large amounts of excess mudstones due to the enormous tunnel excavations and slope cuts would be deposited as landfills.Assessing the deformation and permeability of Yanji mudstone became important for the design,construction and operation of the landfills.This paper presents an experimental study on the deformation and permeability of Yanji mudstone by carrying out a series of oedometer tests with loading/unloading cycles.The results show that the sample with a lower initial water content exhibited greater swelling deformation after inundation,a lower yield stress,greater deformation and a higher hydraulic conductivity during the loading/unloading cycles.As the number of loading/unloading cycles increased,the yield stress and accumulated plastic deformation increased,while the compression index,rebound index and hydraulic conductivity decreased.The samples became stiffer and their hydromechanical behaviour tended to be stable after three cycles.The compression curves could be divided into pre-yield and post-yield zones.The post-yield zones of compression curves and the rebound curves could be normalized into a unique line,and the pre-yield zones of the compression curves could be described as lines.Basic equations were developed to predict mudstone deformation under cyclic loading and unloading.Additionally,an empirical relationship between the hydraulic conductivity and void ratio was also proposed.The ability of the proposed methods was verified by the overall good agreement between the experimental results and predicted values.
文摘The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.
文摘A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper (DOI:10.4236/jmp.2018.910125) we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S(<em>m</em>) and K(<em>m</em>, <em>a</em>) are depending on constant parameters in such a way that S <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0 and K<span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span><span style="white-space:nowrap;"><span style="white-space:nowrap;"></span></span> S when <em>a</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span></span> 0, the CC of S do not provide the CC of M when <em>m</em> <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0 while the CC of K do not provide the CC of S when a <span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">→</span></span></span> 0. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the <em>prolongation/projection</em> (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebra and that the role played by the Spencer operator is crucial. We get K < S < M with 2 < 4 < 10 for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even if each Spencer sequence is isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by the corresponding Lie algebra.
文摘When a differential field <em>K</em> having <em>n</em> commuting derivations is given together with two finitely generated differential extensions <em>L</em> and <em>M</em> of <em>K</em>, an important problem in differential algebra is to exhibit a common differential extension <em>N</em> in order to define the new differential extensions <em>L</em><span style="white-space:nowrap;"><span style="white-space:nowrap;">∩</span></span><em>M </em>and the smallest differential field <span style="white-space:nowrap;">(<em>L</em>,<em>M</em> ) <span style="white-space:nowrap;"><span style="white-space:nowrap;">⊂</span></span> <em>N</em></span> containing both <em>L</em> and <em>M</em>. Such a result allows to generalize the use of complex numbers in classical algebra. Having now two finitely generated differential modules<em> L</em> and <em>M</em> over the non-commutative ring <span style="white-space:nowrap;"><em>D</em> = <em>K </em>[<em>d</em><sub>1</sub>,...,<em>d</em><sub>n</sub>] = <em>K</em> [<em>d</em>]</span> of differential operators with coefficients in <em>K</em>, we may similarly look for a differential module <em>N</em> containing both <em>L</em> and <em>M </em>in order to define <span style="white-space:nowrap;"><em>L</em>∩<em>M</em></span> and <span style="white-space:nowrap;"><em>L</em>+<em>M</em></span>. This is <em>exactly</em> the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a <em>built-in</em> property of a control system, not depending on the choice of inputs and outputs. The purpose of this paper is thus to revisit control theory by showing the specific importance of the two previous problems and the part plaid by <em>N</em> in both cases for the parametrization of the control system. An important tool will be the study of <em>differential correspondence</em><em>s</em>, a modern name for what was called <em>B<span style="white-space:nowrap;"><span style="white-space:nowrap;">ä</span></span>cklund problem</em> during the last century, namely the elimination theory for groups of variables among systems of linear or nonlinear OD or PD equations. The main difficulty is to revisit <em>differential homological algebra</em> by using noncommutative localization as a way to generalize the symbolic calculus in the style of Heaviside and Mikusinski. Finally, when <em>M</em> is a <em>D</em>-module, this paper is using for the first time the fact that the system <span style="white-space:nowrap;"><em>R</em> = <em>hom<sub>K</sub></em> (<em>M</em>,<em>K</em>)</span> is a <em>D</em>-module for the Spencer operator acting on sections, avoiding thus behaviours, trajectories and signal spaces in a purely formal way, contrary to a few recent works on this difficult subject.
文摘When <em>D</em> is a linear partial differential operator of any order, a <em>direct problem</em> is to look for an operator <em>D</em><sub>1</sub> generating the <em>compatibility conditions </em>(CC) <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub><em>1</em></sub><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span> =</span><sub></sub> 0 of <em>D</em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">ξ </span></em></span></span>= <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em><span style="white-space:nowrap;">η</span></em></span></span>. Conversely, when <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is given, an <em>inverse problem</em> is to look for an operator <span style="white-space:normal;"><em>D</em></span> such that its CC are generated by <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> and we shall say that <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>1</sub></span> is <em>parametrized</em> by <em>D</em> = <span style="white-space:normal;"><em>D</em></span><span style="white-space:normal;"><sub>0</sub></span>. We may thus construct a differential sequence with successive operators <em>D</em>, <em>D</em><sub>1</sub>, <em>D</em><sub>2</sub>, ..., each operator parametrizing the next one. Introducing the<em> formal adjoint ad</em>() of an operator, we have <img src="Edit_ecbb631c-2896-4dad-8234-cacd5504f138.png" alt="" />but <span style="white-space:nowrap;"><em>ad</em> (<em>D</em><sub><em>i</em>-1</sub>)</span> may not generate <em>all</em> the CC of <em>ad </em>(<em>D</em><sub>i</sub>). When <em>D </em>= <em>K</em> [d<sub>1</sub>, ..., d<sub>n</sub>] = <em>K </em>[<em>d</em>] is the (non-commutative) ring of differential operators with coefficients in a differential field <em>K</em>, then <em>D</em> gives rise by residue to a <em>differential module M</em> over<em> D</em> while <em>a</em><em style="white-space:normal;">d </em><span style="white-space:normal;">(</span><em style="white-space:normal;">D</em><span style="white-space:normal;">)</span> gives rise to a differential module <em>N =ad (M)</em> over <em>D</em>. The <em>differential extension modules</em> <img src="Edit_55629608-629e-4b52-ac8f-52470473af77.png" alt="" /> with <span style="white-space:nowrap;"><em>ext<span style="font-size:10px;"><sup>0</sup></span></em><em>(M) = hom</em><sub><em>D</em></sub><em> (M, D)</em></span> only depend on <em>M</em> and are measuring the above gaps, <em>independently of the previous differential sequence</em>, in such a way that <span style="white-space:nowrap;"><em>ext</em><sup><em>1</em></sup><em> (N) = t (M)</em> </span> is the torsion submodule of <em>M</em>. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension <em>n</em> and to prove for the first time that the extension modules highly depend on the Vessiot <em>structure constants c</em>. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint <span style="white-space:nowrap;"><em>Beltrami = ad (Riemann)</em></span> which can indeed be parametrized by the operator <span style="white-space:nowrap;"><em>Lanczos = ad (Bianchi) </em></span>in arbitrary dimension, “<em>one step further on to the right</em>” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when <em>D</em> is a Lie operator of finite type, we shall prove that <img src="Edit_3a20593a-fffe-4a20-a041-2c6bb9738d5d.png" alt="" />. It will follow that the <em>Riemann-Lanczos </em>and <em>Weyl-Lanczos</em> problems just amount to prove such a result for <em>i </em>= 1,2 and arbitrary <em>n</em> when <em>D</em> is the <em>classical or conformal Killing</em> operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.
