Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 November 2018 doi:10.20944/preprints201811.0598.v1 The Sun's work 1 Some second-order tensor calculus identities and applications in continuum mechanics Bohua Sunyz, Cape Peninsula University of Technology, Cape Town, South Africa (Received 25 November 2018) To extend the calculation power of tensor analysis, we introduce four new definition of tensor calculations. Some useful tensor identities have been proved. We demonstrate the application of the tensor identities in continuum mechanics: momentum conservation law and deformation superposition. Key words: tensor, gradient, divergence, curl, elasto-plastic deformation Nomenclature a; b; c; d, vector in lower case; A; B, tensor in capital case; X; Y , position vector in reference (undeformed) state; x; y, position vector in current (deformed) state; v, velocity; ek, base vector; Gk, base vector in undeformed state; gk, base vector in deformed state; F , deformation gradient; F e, deformation gradient; F p, deformation gradient; r, gradient nabla; rX , gradient nabla respect to X; rx, gradient nabla respect to x; ", permutation tensor; δij , Kronecker delta; I = δij ei ⊗ ej , unit tensor; ·, dot product; ×, cross product; ⊗, tensor product; σ, Cauchy stress tensor defined in current configuration; τ , Kirchhoff stress tensor defined in current configuration; P , the first Piola-Kirchhoff stress tensor; S, the second Piola-Kirchhoff stress tensor; T , dislocation density tensor; J = det(F ), the Jacobian of deformation gradient F ; ρ, mass density in current configuration; ρR, mass density in reference configuration; f, body force in current configuration; fR, body force in reference configuration. y Currently in Cape Peninsula University of Technology z Email address for correspondence:
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