We study dynamic antiplane cracks in the time domain by the boundary integral equation method (BIEM) based on the integral equation for displacement discontinuity (or crack opening displacement, COD) as a function of stress on the crack. This displacement discontinuity formulation presents the advantage, with respect to methods developed by Das and others in seismology, that it has to be solved only inside the crack. This BIEM is, however, difficult to implement numerically because of the hypersingularity of the kernel of the integral equation. Hence it is rewritten into a weakly singular form using a regularization technique proposed by Bonnet. The first step, following a method due to Sladek and Sladek, consists in converting the hypersingular integral equation for the displacement discontinuity into an integral equation for the displacement discontinuity and its tangential derivatives (dislocation density distribution); the latter involves a Cauchy type singular kernel. The second step is based on the observation that the hypersingularity is related to the static component of the kernel; the static singularity is then isolated and can be expressed in terms of weakly singular integrals using a result due to Bonnet. Although numerical applications discussed in this paper are all for the antiplane problem, the technique can be applied as well to in-plane crack dynamics.
The BIEM is implemented numerically using continuous linear space-time base functions to model the COD on the crack. In the present scheme the COD gradient interpolation is discontinuous at the element nodes while the integral equations are collocated at the element midpoints. This leads to an overdetermined discrete problem which is solved by standard least-squares methods. We use the dynamic BIEM to study a set of problems that appear in earthquake source dynamics, including the spontaneous dynamic crack propagation for a very simple rupture criterion. The numerical results compare favorably with the few exact solutions that are available. Then we demonstrate that difficulties experienced with finite difference simulations of spontaneous crack dynamics can be removed with the use of BIEM. The results are improved by the use of singular crack tip elements. 相似文献
A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed. 相似文献
Mining activities provide a good source of minerals of different nature. On the one hand, the primary minerals for whose formation a geological time-scale is required. On the other hand, secondary minerals, formed from removed products after the earlier weathering and alteration states. These are characteristic of the local geology and the environment context that commonly appears due to the low chemical stability of their original primary minerals. This work shows how quickly the reactions promoting secondary minerals may have taken place, due to the fact that these were found in newly formed solid materials called efflorescences. To achieve this purpose, the sampling is crucial. It was carried out in such a way that tried to guarantee that the samples collected consisted in the very top soil matter (first 2 cm depth). Thus, unlike the deeper soil, the material analysed may have been newly formed due to the interactions that they had with the place weathering agents (i.e. air oxygen, humidity, and microbial activities). Raman spectroscopy has emerged as a good and fast non-destructive technique that provides molecular information of the local mineralogy without the need of any pre-treatment of the samples. At the same time, the work looked for information on the variety of non-stable lead and-or zinc containing minerals due to the possible health and environmental risks they convey. Among the different minerals identified, 16 were of primary nature while 23 may be classified as secondary minerals, probably formed in the last decades as the result of the extractive activities. 相似文献
The dynamic electronic structure of atoms and molecules can be directly observed by means of the (e, 2e) reaction, which measures the distribution of energies and momenta of two electrons in coincidence after a knockout reaction initiated by an electron beam of known momentum incident on a molecular gas target. The molecular state for each event is identified by the electron separation energy. The recoil momentum for each event is known from the difference of measured initial and final momenta. It has been verified that values of this momentum are equal under suitable conditions to the momentum of the electron in the target immediately before knockout. Thus the spherically-averaged electron momentum distribution for each molecular orbital is measured. This is directly related to molecular orbitals calculated by the methods of quantum chemistry. Properties obtained by this method for different types of molecules are discussed. 相似文献
The structures of the modulated phases of K2SeO4, Rb2ZnCl4, Rb2ZnCl4 and Cs2CdBr4 are compared. Both commensurate and incommensurate modulations are considered. Clear common features can be derived between the polarization vectors of their primary harmonic modulations. Even distortions with very different modulation wavevectors present a clear correlation. 相似文献
We use computer algebra to determine all the multilinear polynomial identities of degree ≤7 satisfied by the trilinear operations (a·b)·c and a·(b·c) in the free dendriform dialgebra, where a·b is the pre-Lie or the pre-Jordan product. For the pre-Lie triple products, we obtain one identity in degree 3, and three independent identities in degree 5, and we show that every identity in degree 7 follows from the identities of lower degree. For the pre-Jordan triple products, there are no identities in degree 3, five independent identities in degree 5, and ten independent irreducible identities in degree 7. Our methods involve linear algebra on large matrices over finite fields, and the representation theory of the symmetric group. 相似文献
We introduce degree n Sabinin algebras, which are defined by the polynomial identities up to degree n in a Sabinin algebra. Degree 4 Sabinin algebras can be characterized by the polynomial identities satisfied by the commutator, associator, and two quaternators in the free nonassociative algebra. We consider these operations in a free power associative algebra and show that one of the quaternators is redundant. The resulting algebras provide the natural structure on the tangent space at the identity element of an analytic loop for which all local loops satisfy monoassociativity, a2a ≡ aa2. These algebras are the next step beyond Lie, Malcev, and Bol algebras. We also present an identity of degree 5 which is satisfied by these three operations but which is not implied by the identities of lower degree. 相似文献
Analytical and Bioanalytical Chemistry - The application of a non-destructive analytical procedure to characterise the mineral phases in meteorites is a key issue in order to preserve this type of... 相似文献