PSI - Issue 43

Jiří Tomáš / Procedia Structural Integrity 43 (2023) 65– 70

67 3

J. Toma´sˇ / Structural Integrity Procedia 00 (2023) 000–0 0

Fig. 1. Visualization of a Mo¨bius crystal, using the MATLAB function plot3 . Consider an n -dimensional submanifold N in M . In the language of di ff erential geometry, a subset N of M is said to be an n -dimensional submanifold of M if for any X 0 ∈ N there is local map (coordinate system) ( U , ϕ ) such that ϕ ( U ∩ N ) = ϕ ( U ) ∩ ( R n × { 0 } m − n ). Such a local map on M is said to be adopted to N , cf. Fig. 2. The concept of the N -material isomorphism follows from the situation when the response ψ depends only on vectors from T X N and P 12 ( T X 1 N ) = T X 2 N . This can happen e. e. by monitoring the surface stress determined by two vectors of T X N of a layer modelled by N . In formulas, we write ψ ( FP 12 X 1 , v 1 , X 1 , v 2 , X 1 , v 3 , X 1 ) = ψ ( FP 12 X 1 , v 1 , X 1 , v 2 , X 1 ) = ψ ( X 2 , v 1 , X 2 , v 2 , X 2 , v 3 , X 2 ) = ψ ( X 2 , v 1 , v 2 ) where v 1 , X i v 2 , X i are tangent to N at X i ( i = 1 , 2) and P 12 acts as a TN -preserving tangent map on TM (for the concept of the tangent bundle see Subsection 1.2). If X 1 = X 2 = X , we speak about an N -material symmetry at X . Analogously as in case of ordinary material isomorphisms, place M to R 3 by means of a reference configuration κ 0 adopted to N . Take the so-called N -archetype X 0 ∈ N with the two (or one) tangent vectors of the basis T X 0 M tangent to N . By N -implants we mean N -material isomorphisms P ( X ) between X 0 and X . Consider a linear map µ : T X 0 M → T 0 R 3 R 3 adopted to N . Then P ( X ) ◦ µ − 1 can be corestricted to N or TN and we obtain an element j 1 0 ϕ ∈ reg J 1 0 ( R 3 , N ) X . By reg we indicate elements ϕ : R 3 → N with the maximal rank of jacobian. We remark that in general, j 1 X f is identified with the tangent map T X f at X , which is by the way linear and can be considered as intrinsically defined multiple-value gradient. Applying coordinates we come to the space reg J 1 0 ( R 3 , R n ) 0 , n = 1 , 2 or for general parameters to reg J 1 0 ( R k , R n ) 0 , k ≥ n . Applying the so-called R n -material symmetries at 0 ∈ R n modelled by the left action of the linear group G 1 n = inv J 1 0 ( R n , R n ) 0 = GL( n ) or by its subgroup on reg J 1 0 ( R 3 , R n ) 0 and taking the orbits with respect to such left action we obtain the classical Grassmannian Gr ( k , m ), cf. Fig. 2. The more exact argument to the recent coordinate step will be given by the functionality of Grassmannian in the very end of the paper. Roughly speaking, for m ≤ k , Gr ( k , m ) is the typical or standard object formed by orbits of the left action of the linear group GL( m ) on reg J 1 0 ( R k , R m ) 0 such that composing the representative elements of such orbits with elements from reg J 1 0 ( R m , M ) X from the left yields possible M -implants at points X ∈ M and consequently M -uniform fields, remarking that M is an m -dimensional manifold considered mostly as an m -dimensional submanifold of R k . In the very end of the present subsection we state that the classical Grassmannian Gr ( k , m ) is defined as the space of all m -dimensional linear subspaces of R k . For n = 1 we have the well-known projective space, which is the basis of the so-called line bundle. 1.2 Basic geometrical concepts . We follow the geometrical terminology from Kola´ˇr et al (1993) and Sharpe (1997). For a smooth curve γ : R → M , an r -jet j r t 0 γ at t 0 ∈ R is defined as the equivalence class of all curves having the r -th order contact at x = γ ( t 0 ) ∈ M in the sense of the same value of derivatives of the function ϕ ◦ γ at t 0 up to order r for any function ϕ on M defined near x = γ ( t 0 ). For a smooth map f : M → N of manifolds, the r -jet j r x f is defined by the assignment j r 0 γ → j r 0 ( f ◦ γ ). The points x and y are said to be the source and target of the jet Y = j r x f , which can be written by α ( Y ) = x and β ( Y ) = f ( x ). The concept of the r -jet intrinsically expresses the system of r -th order dim N -valued Taylor polynomials in various coordinate systems centred at x . For any Y = j r x f and Z = j r f ( x ) g there is defined the composition Z ◦ Y by j r x ( g ◦ f ), which is correct, i. e. independent on the choice of f ∈ Y and g ∈ Z . The space of all elements of this kind is denoted by J r ( M , N ) or J r x ( M , N ) et. depending on free or fixed source or target. In terms of 1-jets we define the tangent bundle TM on a manifold M by T X M = { j 1 0 γ ; γ (0) = X } , TM = x ∈ M T X M . For a smooth map f : M → N we define T f : TM → TN by j 1 0 γ → j 1 0 ( f ◦ γ ). Further, there is the tangent bundle