计算地震学Introduction宣讲培训.ppt

;;;;;Question: There are 9 unknowns (3 displacement components + 6 stress components), but only 3 differential equations of motion. Other 6 equations or constraints are needed, that are given by the material law. 1.3 Constitutive relations Hooke’s law In general, a linear elastic medium is described by ?ij = Cijkl ekl where Cijkl are elastic constants (tensor of order 4) and independent from the stress state. Symmetry of Cijkl (that reduces the number of C from 81 to 21): (1) Cjikl = Cijkl, (from ?ji = ?ij) (2) Cijlk = Cijkl, (from elk = ekl) (3) Cijkl = Cklij, (from the 1. thermodynamic law) In case of isotropy, it includes only two Lame’s constants Cijkl = ? ?ij ?kl + ? ( ?ik ?jl + ?il ?jk ) Hooke’s law: ?ij = ? ekk ?ij + 2? eij = ? ukk ?ij + ? (ui,j + uj,i) (that will be mostly used!);1.4 Boundary-value problems for a self-gravitating and hydrostatically pre-stressed Earth Initial equilibrium ? po(x) = ?o(x) ? ?o(x) ?2 ?o(x) = –4?G ?o(x) where ?o = density, ?o = gravitational potential (go = ? ?o), po = hydrostatic pressure, G = 6.67?10-11 m3kg-1sec-2 (gravitational constant). The index o denotes the unperturbed initial state. Description of displacement of a continuum u = uL(x,t) = r(x,t) – x --- Lagrangeian (material) description = uE(r,t) = r – x(r,t) --- Eulerian (local) description Lagrangian description: One observes the change in the vicinity of a selected particle, i.e. material-fixed Eulerian description: One observes the change in the vicinity of a selected location, i.e. space-fixed;;;;;;;;;;;;;;;;;;;;;;;;;;3.2 Fundamental solutions The eigenvalue and eigenvector problem In a source-free region, the homogeneous equations of motion is used: ?z y(z) = A y(z), ( zn zn+1, n ? s) The general form of solutions y(z) = a e?z ? ( ?I – A ) a = 0 ? det (