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851.
852.
We consider an elastic plate with the non-deformed shape ΩΣ := Ω \ Σ, where Ω is a domain bounded by a smooth closed curve Γ and Σ ⊂ Ω is a curve with the end points {γ1, γ2}. If the force g is given on the part ΓN of Γ, the displacement u is fixed on ΓD := Γ \ ΓN and the body force f is given in Ω, then the displacement vector u(x) = (u1(x), u2(x)) has unbounded derivatives (stress singularities) near γk, k = 1, 2 u(x) = ∑2k, l=1 Kl(γk)r1/2kSCkl(θk) + uR(x) near γk. Here (rk, θk) denote local curvilinear polar co-ordinates near γk, k = 1, 2, SCkl (θk) are smooth functions defined on [−π, π] and uR(x) ∈ {H2(near γk)}2. The constants Kl(γk), l = 1, 2, which are called the stress intensity factors at γk (abbr. SIFs), are important parameters in fracture mechanics. We notice that the stress intensity factors Kl(γk) (l = 1, 2; k = 1, 2) are functionals Kl(γk) = Kl(γk; ℒ︁, Ω, Σ) depending on the load ℒ︁, the shape of the plate Ω and the shape of the crack Σ. We say that the crack Σ is safe, if Kl(γk; Ω)2 + K2(γk; Ω)2 < RẼ. By a small change of Ω the shape Σ can change to a dangerous one, i.e. we have K1(γk; Ω)2 + K2(γk; Ω)2 ⩾ RẼ. Therefore it is important to know how Kl(γk) depends on the shape of Ω. For this reason, we calculate the Gâteaux derivative of Kl(γk) under a class of domain perturbations which includes the approximation of domains by polygonal domains and the Hadamard's parametrization Γ(τ) := {x + τϕ(x)n(x); x ∈ Γ}, where ϕ is a function on Γ and n is the outward unit normal on Γ. The calculations are quite delicate because of the occurrence of additional stress singularities at the collision points {γ3, γ4} = Γ D ∩ Γ N. The result is derived by the combination of the weight function method and the Generalized J-integral technique (abbr. GJ-integral technique). The GJ-integrals have been proposed by the first author in order to express the variation of energy (energy release rate) by extension of a crack in a 3D-elastic body. This paper begins with the weak solution of the crack problem, the weight function representation of SIF's, GJ-integral technique and finish with the shape sensitivity analysis of SIF's. GJ-integral Jω(u; X) is the sum of the P-integral (line integral) Pω(u, X) and the R-integral (area integral) Rω(u, X). With the help of the GJ-integral technique we derive an R-integral expression for the shape derivative of the potential energy which is valid for all displacement fields u ∈ H1. Using the property that the GJ-integral vanishes for all regular fields u ∈ H2 we convert the R-integral expression for the shape derivative to a P-integral expression. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
853.
Marius D. R. Lutz Hongyu Zhong Nils Trapp Bill Morandi 《Helvetica chimica acta》2023,106(4):e202200199
We report the synthesis of coordinatively unsaturated cationic rhodium complexes bearing the sterically encumbered electron-rich NHC ligand IPr*OMe. The COD (1,5-cyclooctadiene) complex [Rh(IPr*OMe)(COD)]BF4 adopts a tilted, pseudo-square planar coordination geometry, where bonding to the ipso-carbon of the NHC aryl substituent was observed in the solid state. Hydrogenation of this complex afforded a metastable dihydride complex [Rh(IPr*OMe)(H)2]BF4 with an unusual internal coordination to an arene of the ligand. In the absence of a hydrogen atmosphere, spontaneous reductive elimination of H2 afforded a rhodium complex [Rh(IPr*OMe)]BF4 with a single chelating ligand that stabilizes the highly unsaturated metal by two-fold π-face donation as suggested by NMR spectroscopy and computational studies. This unusual complex might serve as a versatile precatalyst for a variety of transformations. 相似文献