PSI - Issue 23

Miroslav Kureš / Procedia Structural Integrity 23 (2019) 396–401 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

398

3

and hence ( v 1 , v 2 , v 3 ) = ( ku 1 , ku 2 , ku 3 ) for an arbitrary k  R . Q. E. D. It reads as the bound vectors u ( X ) and u ( Y ) possess the same moment u

0 if and only if X and Y lie on a line with the

direction vector u .

1.3. Motors and screws By a motor we understand a couple ( u , u 0 ) . “Motor” is a combination of words “moment” and “vector” , Dimentberg (1968). It is a representation of a vector system, expressed by the principal vector and principal moment of the system. However, our radius vectors have been so far based on the origin O of the coordinate system, but another point may be such a reference point. Every motor can be brought to such a reference point that its moment and vector parts become colinear, which turns a motor into an equivalent screw . A parallel line through this point is the screw axis , see Brodsky and Shoham (1999). The screw calculus , Dimentberg (1968), is based on the basis of the apparatus of modern vector algebra using dual numbers. Traditional mechanics of continua endows particles of a material body with translational degrees of freedom, the Cosserat brothers' approach endows them with both translational and rotational degrees of freedom. In elementary approach a body B of dimension 1 (rods, beams) or 2 (plates, shells) or 3 in R 3 is considered. For each particle of such a body we consider its initial position (a radius vector) and its initial settings of 3 orthonormal directors. Such approach is based on differential geometry theory applied to mechanics and there is no doubt that Cosserat continuum theory is suitable e.g. for describing the kinematics of granular media. The mathematical description can be based on motors, as stated in the monograph of Vardoulakis (2018) in which the basic theorems used to formulate the Cosserat continuum, together with the appropriate kinematic fields conjugate to the motor vectors; kinematic motors are compound vectors including linear velocity and spin (angular velocity), fully describing a rigid body motion in the new reduced geometric representation. 2.1. Dual vectors and dual curves The dual numbers are defined as numbers of a form a = a + A ; a, A  R , which extend the real numbers by adjoining new (“infinitesimal” ) element with the property 2 = 0. The set of dual numbers is denoted by D and it forms a two dimensional commutative unital associative R-algebra. The arithmetic of dual numbers has several specifics, so we refer e.g. to the paper Kure š (to appear) for details. Further, a dual function f : D  D of a dual variable x = x + X can be represented in the form ( + ε) = φ( , ) + Φ( , ) Dual functions which are also differentiable in a neighborhood of a point are called synektic . This is satisfied if and only if ∂ ∂ φ = ∂ ∂ Φ and = 0. It is an analogy with the analytic functions over C which comply with the Cauchy-Riemann conditions. Then we have 1.4. Application: Cosserat media 2. Dual curves