By Alexander Konyukhov
This ebook incorporates a systematical research of geometrical events resulting in touch pairs -- point-to-surface, surface-to-surface, point-to-curve, curve-to-curve and curve-to-surface. every one touch pair is inherited with a different coordinate procedure in line with its geometrical homes similar to a Gaussian floor coordinate procedure or a Serret-Frenet curve coordinate method. The formula in a covariant shape permits in a simple type to think about quite a few constitutive kinfolk for a undeniable pair similar to anisotropy for either frictional and structural components. Then normal equipment renowned in computational touch mechanics resembling penalty, Lagrange multiplier tools, 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 pursuits of such modifications are then ready-for-implementation numerical algorithms in the finite point technique together with any arbitrary discretization options similar to excessive order and isogeometric finite components, that are such a lot handy for the thought of geometrical situation.
The publication proposes a constant research of geometry and kinematics, variational formulations, constitutive kin for surfaces and discretization thoughts for all thought of geometrical pairs and includes the linked numerical research in addition to a few new analytical leads to touch mechanics.
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Extra resources for Computational Contact Mechanics: Geometrically Exact Theory for Arbitrary Shaped Bodies
A contact description of many geometrical features (curve-to-curve, curveto-surface) is almost not possible because of the necessity of convective surface coordinates. • Geometrically motivated measures of contact interaction are coupled with convective variables in a specially deﬁned coordinate system. This straightforwardly leads to a description via convective variables. 4 Goals and Structure of the Book 17 geometrical objects such as surfaces, edges etc. and can be handled – mostly improperly – with great eﬀorts.
As a result a contact layer element allowing anisotropic p−reﬁnement is created. The layer contact element is applied then to initially linear meshes. A good correlation with the analytical Hertz problem is achieved even within a single contact layer element. Chapter 11 is specially devoted to the extensive numerical analysis of the coupled anisotropic adhesion-friction model developed in Chapter 6. 4 Goals and Structure of the Book 23 coordinate system and a spiral orthotropy of a cylindrical surface.
33) The term ∇j T i is a covariant derivative of the contravariant component T i ∇j T i = ∂T i i + T k Γjk . 34) A similar expression can be found for the covariant derivative of the covariant components Ti : ∂Ti ∇j Ti = j − Tk Γijk . 35) ∂ξ In the case of covariant components we need the derivative of a contravariant ∂ρ i base vector instead of ρij , see eqn. 32). 37) and the covariant derivative for the covariant component gets the following form ∂Ti ∇j Ti = j − Tk Γijk . 2 Deﬁnition of the Curve in 3D and Its Geometrical Characteristics Let us consider a curve in 3D Cartesian space arbitrarily parametrized with a parameter ξ (1D manifold).