PSI - Issue 43

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

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

Fig. 2. A simple case of G (2 , 1) and its local sections (left-hand-side scheme). Curve as a submanifold of surface (right-hand-side scheme). projection p M : TM → TN satisfying p N ◦ T f = f ◦ p M . Smooth maps ξ M : M → TM satisfying p M ◦ ξ M = id M , i.e. local sections of p M are said to be vector fields on M . There is the jet functor J r (Kola´ˇr et al (1993), Sections 12 and 18) assigning J r ( M , N ) to any couple of manifolds M , N , dim M = m and the map J r ( g , h ) : J r ( M 1 , N 1 ) → J r ( M 2 , N 2 ) defined by j r x f → j r f ( x ) h ◦ j r x f ◦ ( j r x g ) − 1 to any couple of maps g , h where g is a local di ff eomorphism between M 1 and M 2 and h : N 1 → N 2 is arbitrary. The jet group G r k is defined as inv J r 0 ( R k , R k ) 0 with the multiplication defined by the composition of r -jets. There are the higher-order velocity bundle functors T r k defined by T r k M = J r 0 ( R k , M ) and T r k f = J r 0 (id R k , f ) for any smooth map f : M → N . Further, there is a frame bundle functor P r defined by P r M = inv J r 0 ( R m , M ) on objects and by P r f = J r 0 (id R m , f ) on morphisms. The mechanical counterparts of frames, their morphisms and the right action are implants, material isomorphisms and changes of archetypes. The right action of the jet group G r m yields the structure of a principal bundle on P r M , see Kola´ˇr et al (1993), Section 12, or Sharpe (1997), Section 5. As for the classical (first-order) Grassmannian Gr ( k , m ), its manifold structure is usually defined in terms of the homogeneous space O ( k ) / ( O ( m ) × O ( k − m )) in the sense of the Lie theory (Kola´ˇr et al (1993), Sections 5, 10 and Sharpe (1997), Chapter 5, Section 2 for more detailed version) where O indicates the orthogonal group. It is a space of left cosets of the group O ( m ) × O ( k − m ) acting from the right on O ( k ). If we do not accent the orthogonality, Gr ( k , m ) can be defined as the space of right cosets G 1 k , m \ G 1 k , which can be identified with the space of of G 1 m = GL( m ) (see Sharpe (1997), Chapter 5) acting from the left on reg J 1 0 ( R k , R m ) 0 . In other words, Gr ( k , m ) consists of orbits of bases of m -dimensional linear subspaces under linear automorphisms corresponding to elements from G 1 m = GL( m ). Higher-order and Weil Grassmannians are introduced and disscussed in Section 3.

2. Higher-order and Cosserat models 2.1 Higher-order model . Instead F = grad χ = grad( κ ◦ κ − 1 the r -th order. Fixing the reference configuration κ 0 , the constitutive equation ψ = ψ ( j r Then the definition formula for material isomorphism P 12 = j r X 1 p 12 reads ψ ( j r X 2 speak about material symmetries. They constitute the symmetry group G X at X . Select an archetype X 0 with j r 0 µ − 1 for regular µ : M X 0 → R 3 κ ◦ j r

0 ) applied in the first order case consider j r κ 0 ( X )

χ in case of

κ ( X ) χ ) is identified to ψ = ψ ( j r

X κ ).

= ψ ( j r

X 1 p 12 )

κ ). If X 1 = X 2 we

X 2

0 mapping X 0 to 0. Then material isomorphisms

P ( X ) = j r X 0 p ( X ) from X 0 to X are said to be implants. Any P ( X ) can be identified with P ( X ) ◦ j r 0 µ − 1 : T r 0 R 3 → T r X M , i.e. with the r − th -order frame. Thus P ( X ) ∈ P r X M . A collection of all implants P ( X ) assigned smoothly to points X ∈ M is said to be a uniformity field. A change of an archetype corresponds to the assignment P ( X ) → P ( X ) ◦ ( j r 0 µ ◦ ( j r 0 µ ) − 1 ), the last expression in brackets being an element of jet group G r 3 and represents the right action in the definition of the higher-order frame bundle. Given a uniformity field, the equation ψ = ψ ( j r κ 0 ( X ) χ, X ) ψ ( j r X κ ) can be replaced by ψ = ¯ ψ ( j r X κ ◦ j r 0 p X ), where elements of the form j r 0 p X = P ( X ) represent implants. Making analogous deductions as in the first-order case we come to the r -th order material G -structure, which is a reduction of the higher-order frame bundle to the subgroup G of G r m (in mechanics mostly m = 3). This group corresponds to the system of conjugated groups G X . Analogously as in the first order case we can do with the r -th order Grassmannians as an abstract object for the description of higher-order N -material isomorphisms, N symmetries, N -implants etc.