PSI - Issue 43

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

66

2

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

κ the current one. Hyperelasticity requires an additional claim of the only one constitutive response expressed by a scalar quantity ψ . Geometrical concepts and theories are adopted from the books Kola´ˇr et al (1993) and Sharpe (1997). A material body B is modelled by a 3-dimensional manifold, which we denote rather M due to the prevailing mathematical character. Further, there a reference configuration κ 0 , an embedding of M to the physical space E 3 endowed with the standard inner product and metrics. Other configurations κ : M → R 3 may be considered with ranges in the a ffi ne space only, but the metrics can be transmitted from the range of κ 0 declaring some curvilinear coordinate system to be orthogonal. As a matter of fact, configurations coincide with local charts on a manifold. For an elastic first-order material, a material isomorphism between points X 1 and X 2 is defined as a linear map P 12 : T X 1 M → T X 2 M with respect to which the constitutive response is invariant. A material isomorphism is defined in the reference configuration and by F applied from the left it is transmitted to other configurations. In case of hyperelasticity its definition condition reads ψ ( FP 12 , X 1 ) = ψ ( F , X 2 ). Geometrically, P 12 corresponds to the tangent map T X 1 f for some local di ff eomorphism f over M defined near X 1 . If all points are mutually materially isomorphic we speak about a uniform body. In case of X 1 = X 2 = X , we speak about a material symmetry at X . Material symmetries at X form a group G X . For the set P 12 of all material isomorphisms above, it is easy to see that for any P 12 it holds P 12 = P 12 ◦ G 1 = G 2 ◦ P 12 and G 2 = P 12 ◦ G 1 ◦ P − 1 12 (the so-called Noll rule), by G i denoting G X i . Consider a uniform body M and place it to R 3 by means of the reference configuration κ 0 . Select some of its points X 0 as the so-called archetype, which is particularly a material point. Take the regular linear map µ : T X 0 M → T 0 R 3 R 3 , which is in the sequel the tangent map T f for some map f : M → R 3 . By an implant we mean a material isomorphism P ( X ) between X 0 and X . Any implant P = P ( X ) can be identified with P ( X ) ◦ µ − 1 : T 0 R 3 R 3 → T X M which is (the first-order) frame. Recall the classical geometrical concept of the frame as a linear map assigning to a selected basis of the tangent space T 0 R 3 R 3 at zero (mostly the canonical basis) a basis of the tangent space T X M . In terms of jets (see Subsection 1.2) we define for a manifold M and a local di ff eomorphism f : M → N the object P 1 M = inv J 1 0 ( R m , M ) and P 1 f : P 1 M → P 1 N as the assignment j 1 0 ϕ → j 1 0 ( f ◦ ϕ ) (see Subsection 1.2). Clearly, any frame on M can be composed with material isomorphisms on M , generating another implant. In particular, implants can be composed with elements of the symmetry group G X . Changing µ to µ yields P = P ◦ ( µ ◦ µ − 1 ), which is the value of the free right action of the element j 1 0 ( µ ◦ µ − 1 ) from the jet group G 1 3 = J 1 0 ( R 3 , R 3 ) 0 acting on P ∈ J 1 0 ( R m , R m ) (see Subsection 1.2 below). Finally, we have (the first-order) material G-structure , which is a reduction of the principal bundle P 1 M → M to the subgroup G of the linear group GL(3) with the principal right action of G given by its composition with elements of P 1 M from the right. We remark that the left action of G X on fibers over points X ∈ M is given by the jet composition from the left and G X are conjugated to G . For details concerning the concept of the principle bundle see Kola´ˇr et al (1993), Section 10 or Sharpe (1997), Section 5. We recall the concept of the uniform field from Epstein et al (2007), which assigns smoothly an implant at X to any X ∈ M , at least locally in case of an only locally uniform body M . We remark that in case of a trivial or discrete symmetry group G , the uniform field is uniquely determined by an archetype. In case of a continuous symmetry group it not so, there is a freedom in the choice governed by G . Example . We demonstrate the concepts of material isomorphism, material symmetry, local and global uniformity field on the so-called Mo¨ebius crystal, see Wang (1967), Wang and Truedell (1973). Mo¨bius crystal arises from the cylinder of length l and radius r satisfying R = l / 2 π > r by twisting the cylinder by π = 180 ◦ and welding it on the ends. Consider the rhombic symmetry, i. e. symmetries by 180 ◦ to each of the coordinate axes. In such case we have obviously a discrete symmetry group. In the initial state, material isomorphisms correspond exactly to translations, inducing the global uniformity fields. Leaving the initial state, material points are equally twisted along the longitudinal axis of the cylinder up to 180 ◦ and we obtain the non-trivial material isomorphisms from [ x 0 , y 0 , z 0 ] = [ R + r cos ϕ, r sin ϕ, 0] to [ x , y , z ] = ( R + r cos( ϕ + ψ/ 2)) cos ψ, r sin( ϕ + ψ/ 2) , ( R + r cos( ϕ + ψ/ 2)) sin ψ ] for ϕ ∈ 0 , 2 π and ψ ∈ 0 , 2 π , see Fig. 1 The uniformity field we have obtained is not global (there is the 180 ◦ -leap), it is only local. On the other hand, suppose that the body is at least transversely isotropic about the longitudinal axis (i. e. any rotation about that axis is a material symmetry). In this case we have obviously a continuous symmetry group. Then all of the Euclidean translations are material isomorphisms again and if we weld the cylinder to the torus we obtain a globally smooth field of material ismorphisms.