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

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2.2 Cosserat model . In the original works of Cosserat brothers there was the intention to model microlinear struc tures with internal degrees of freedom, first with any point being endowed with a rigid triad of orthogonal vectors. Emulating gradually the model mentioned above on comes to materials corresponding intuitively to the standard body like concrete (formed by a cement matrix) augmented by a densely distributed collection of grains. In the limit, each point of the matrix is carrying a grain whose deformation is considered in the first approximation as a homo geneous one due to the smallness. In this sense, Cosserat body (of the first order) corresponds to the frame bundle π M : P 1 M → M with the underlying body M as a macromedium or matrix and the point of P 1 M themselves as grains over the stones from the basis, which are carries of information about events of the grain modelled by a ffi ne deformations. The points of P 1 M are endowed with 1-jets of configurations K : P 1 M → P 1 R 3 in the form of principal bundle morphisms. Fibres π − 1 ( X ) consisting of grains over ”stones” X ∈ M are carriers of information about events on the ”grain” level expressed by an a ffi ne transformation. By R we denote the right action of GL(3). P r M K P r R 3 P r M K P r R 3 = X 1 = X , we speak about symmetries at X . They constitute the group G X . An archetype is a selected point X 0 with a fixed configuration which can be in its inverse identified with the jet of a principal bundle morphism from P 1 0 R 3 to P 1 0 M . Transformation laws for the change of reference configurations and the change of archetype correspond to formulas for second order non-holonomic jets and it is necessary to work with he second-order non-holonomic frames. Implants are of the form j 1 0 p X , where p X are material isomorphisms from X 0 to X , which can be in the same way as in the case of the archetype identified with 1-jets of Cosserat configurations K . Besides classical jets there are also the non-holonomic jets and their spaces J r ( M 1 , M 2 ), Ehresmann (1954), Ep stein et al (2007) defined by induction as follows. For r = 1 we have J 1 ( M 1 , M 2 ) = J 1 ( M 1 , M 2 ) with the ordinary composition of 1-jets. Suppose that J r ( M 1 , M 2 ), J r ( M 2 , M 3 ) are defined as well as the composition of their elements defined by ◦ r . To define non-holonomic ( r + 1)-jets, consider local sections σ : M 1 → J r ( M 1 , M 2 ) defined near x ∈ M 1 . Then elements of J r ( M 1 , M 2 ) are of the form j 1 x σ . To define the composition, assume X 1 = j 1 x σ 1 ∈ J r ( M 1 , M 2 ) and X 2 = j 1 y σ 2 ∈ J r ( M 2 , M 3 ) where y = β ( X 1 ) = α ( X 2 ). Then X 2 ◦ r + 1 X 1 is defined by j 1 x ( σ 2 ( β ( σ 1 ( u )) ◦ r σ 1 ( u )). In problems on connections and inhomogenities in Cosserat model, there also appear semi-holonomic jets. All cases of jets can be studied in terms of the Weil functor theory. From this point of view, semi-holonomic velocities were studied eg. in Kuresˇ (2014). 2.3 Weil functors . Weil functors (Kola´ˇr et al (1993)) are exactly those bundle functors which preserve categorial products, Kainz and Michor (1987). On the other hand, they generalize the higher-order velocity bundles T r k . They are in 1-1 correspondence with Weil algebras, i.e. algebras of the form A = R ⊕ N A where N A is the nilpotent ideal. A Weil algebra can be equivalently defined by A = D r k / I where D r k is the algebra of polynomials of order at most r in k indeterminates with the truncated polynomial multiplication and I is its ideal. Further, width A is defined as dim( N A / N 2 A ). By p A : D r k → A we denote the projection homomorphism and by p A , M : T r k M → T A M the induced projection of Weil functors. For A = D r k / I two r -jets j r 0 g and j r 0 h are said to be I -equivalent i ff j r x γ ◦ j r 0 g − j r x γ ◦ j r 0 h ∈ I for any function γ : M → R defined near x ∈ M . Classes of such equivalence are denoted by j A g and T A M is defined as the space of them. For a smooth map ϕ : M → N we define the map T A ϕ by T A ϕ ( j A g ) : = j A ( ϕ ◦ g ). The 1-1 correspondence between Weil algebras and Weil functors is given by the maps A → T A and F → F R , applying the product preserving property of a Weil functor. Natural transformations correspond to homomorphisms of Weil algebras defined by means of elements of jet groups, Kola´ˇr (1986). Consider the groups Di ff r , M x = inv J r x ( M , M ) x , Di ff r , M ( x , 0) = J r ( x , 0) ( M × R k − m ) ( x , 0) with the subgroup (Di ff r , M k , m ) ( x , 0) pro jectable to Di ff r , M x . For M = R m , x = 0 we have G r m , G r k and G r k , m . For j r 0 α x ∈ P r x M let j r 0 ¯ α x denote j r 0 α x × { j r 0 id R k − m } . We generalize the classical Grassmannian to higher-order and Weil cases. Their introduction is based on advanced geometrical theories, namely those of Lie groups, jet groups, homogeneous spaces and Weil functors. π π R R R M κ R 3 P r M κ P r R 3 corresponding the constitutive equation: ψ = ψ ( j 1 X K ). P 12 = j 1 X 1 p 12 is said to be a material isomorphism if ψ ( j 1 X 2 K ◦ j 1 X 1 p 12 ) = ψ ( j 1 X 2 K ). If X 2