Tensor as a tool in engineering analysis

A.P. Akinola, A.S. Borokinni, O.O. Fadodun

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Abstract. This paper underscores the potency of the invariant character of tensor and its derivative concepts and accentuate the synergy between isotropic tensor and other tensors and the corresponding vector operations. The equivalence of covariant derivative in a curvilinear coordinates system embedded with a non-constant vector field and the partial derivative in an affine coordinates system ingrained with a constant vector field is interrogated. The corresponding role of the Christofell symbols as the affine connector of vectors with their derivatives in a variable field are compared to the Frenet-Seret skew-matrix connecting the trihedrons (i.e. tangent, normal and binormal) of a moving space curve with their derivatives. The nexus of the Christofell symbols with the geodesics is also shown. The structure of the metric tensor and the Levi-Chivita skew-symmetric tensor , as isotropic tensor rank-2 and rank-3 respectively is highlighted, such that the usual operations of dot product (or scalar product or inner product) and cross product or (vector product or spin/rotation operation}) are now expressed through the isotropic tensors. Recalling the theory of exterior differential form and invoking the Poincare’s theorem we show the application of the exterior product in establishing exact differential (or total differential) in calculus in relation to plane problem of Elasticity. The invariant nature of the tensor objects and operations therefrom are then copiously invoked and deployed to establish constitutive relation for materials: in finite elasticity, within the context of hyperelasticity; composites, where there is a trade-off between heterogeneity and anisotropy through homogenisation process whereby differential equations with variable coefficients are converted to differential equations with constant coefficients; and plasticity, where application of tensor is exhibited with strain gradient plasticity, and shown how the concepts provide balance of microscopic forces, balance of macroscopic forces, and plastic flow laws as concise mathematical equations.

Invariants, Christofell Symbols, Exterior Differential Forms, Hyperelasticity, Strain Gradient Plasticity

Published online 8/10/2023, 15 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: A.P. Akinola, A.S. Borokinni, O.O. Fadodun, Tensor as a tool in engineering analysis, Materials Research Proceedings, Vol. 31, pp 422-436, 2023

DOI: https://doi.org/10.21741/9781644902592-44

The article was published as article 44 of the book Advanced Topics in Mechanics of Materials, Structures and Construction

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

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