By Alexander Konyukhov
This ebook incorporates a systematical research of geometrical occasions resulting in touch pairs -- point-to-surface, surface-to-surface, point-to-curve, curve-to-curve and curve-to-surface. each one touch pair is inherited with a unique coordinate procedure in response to its geometrical homes comparable to a Gaussian floor coordinate process or a Serret-Frenet curve coordinate procedure. The formula in a covariant shape permits in an easy model to contemplate a number of constitutive relatives for a definite pair corresponding to anisotropy for either frictional and structural components. Then commonplace tools popular in computational touch mechanics akin to penalty, Lagrange multiplier equipment, blend of either and others are formulated in those coordinate platforms. Such formulations require then the robust equipment of differential geometry of surfaces and curves in addition to of convex research. the ultimate ambitions of such adjustments are then ready-for-implementation numerical algorithms in the finite aspect strategy together with any arbitrary discretization innovations equivalent to excessive order and isogeometric finite components, that are so much handy for the thought of geometrical situation.
The ebook proposes a constant examine of geometry and kinematics, variational formulations, constitutive family members for surfaces and discretization options for all thought of geometrical pairs and comprises the linked numerical research in addition to a few new analytical ends up in touch mechanics.
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Additional info for Computational Contact Mechanics: Geometrically Exact Theory for Arbitrary Shaped Bodies
30 2 Diﬀerential Geometry of Surfaces and Curves Fig. 4 Structure of a surface in the vicinity of the parabolic point. are fulﬁlled for each surface point of the surface, then a ﬂat surface is obtained. g. a cylinder, or a cone) can be unwrapped on a plane. 4) Both k1 = 0 and k2 = 0 — a planar point . The local structure can not be identiﬁed without the analysis of the higher order derivatives. Nevertheless, the case with k1 = 0, k2 = 0 for each point leads to a plane in 3D. Eqn. 21) and the mean curvature H is deﬁned as H= 1 a22 h11 − 2a12 h12 + a11 h22 1 (k1 + k2 ) = .
1, as follows: r(ξ 1 , ξ 2 , ξ 3 ) = ρ(ξ 1 , ξ 2 ) + nξ 3 . 8) All necessary parameters are deﬁned in the covariant basis of the surface ρ1 × ρ2 tangent vectors ρ1 , ρ2 and a unit normal vector n = . 2) will be considered in this coordinate system. The introduction of such a coordinate system is the basis of the covariant description for contact problems, developed in Konyukhov and Schweizerhof ,  for isotropic frictional problems, and then in ,  for anisotropic frictional problems.
4 Structure of a surface in the vicinity of the parabolic point. are fulﬁlled for each surface point of the surface, then a ﬂat surface is obtained. g. a cylinder, or a cone) can be unwrapped on a plane. 4) Both k1 = 0 and k2 = 0 — a planar point . The local structure can not be identiﬁed without the analysis of the higher order derivatives. Nevertheless, the case with k1 = 0, k2 = 0 for each point leads to a plane in 3D. Eqn. 21) and the mean curvature H is deﬁned as H= 1 a22 h11 − 2a12 h12 + a11 h22 1 (k1 + k2 ) = .