Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics

affine connection affine space atlas basis body boundary called Cauchy stress Cauchy-Green Chapter components configuration connection Consider continuum mechanics coordinate representation corresponding covector curvature curve defined definition deformation gradient denoted density derivative diffeomorphism differential dual dx¹ elastic elements embedding equations Euclidean space exterior fiber formula frame function geometric homomorphism isomorphism layers Lie derivative Lin(V linear mappings material points non-Euclidean notion obtains open set open subset operation physical space principal bundle pullback Ra)² reference shape relation represented Riemannian metric s-dimensional scalar Section smooth chart smooth manifold smooth mapping smooth vector spatial stress tensor submanifold Suppose tangent space tangent vector tensor field theorem theory tion topological transformation two-point tensor Vec(M vector bundle vector field vector space velocity Θα φα дх дхі