Geodesic
Reference shape of bodies with enhanced kinematics. Part II. Second gradient and microstructure
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 29 (2023) no. 4, pp. 54-76

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The work develops differential-geometric methods for modeling finite incompatible deformations of hyperelastic solids with enhanced kinematics. The response of such bodies, along with the standard kinematic field represented by the deformation gradient, is characterized by additional tensor fields. As such, the paper considers: 1) the second deformation gradient and 2) the tensor field of the second rank, modeling the microstructure of the body. For each of these two cases, compatibility conditions are obtained and their geometric interpretation is proposed. Geometry is synthesized on the material manifold representing a body with enhanced kinematics. The corresponding affine connection has non-zero torsion and curvature, which can be useful for modeling a body with dislocations and disclinations.
Keywords: hyperelasticity, body with enhanced kinematics, residual stresses, non-Euclidean geometry, material metric, material connection, curvature, non-metricity.
Mots-clés : second gradient, microstructure, incompatible deformations, torsion 
@article{VSGU_2023_29_4_a2,
     author = {K. G. Koifman},
     title = {Reference shape of bodies with enhanced kinematics. {Part} {II.} {Second} gradient and microstructure},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {54--76},
     publisher = {mathdoc},
     volume = {29},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2023_29_4_a2/}
}
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K. G. Koifman. Reference shape of bodies with enhanced kinematics. Part II. Second gradient and microstructure. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 29 (2023) no. 4, pp. 54-76. http://geodesic.mathdoc.fr/item/VSGU_2023_29_4_a2/