Abstract
Continuum mechanics has been formulated mainly in the mathematical framework of tensor algebra and tensor calculus. The accurate understanding and the proper application of continuum damage mechanics, therefore, necessitate sound foundation of this mathematical subject. The present chapter is the presentation of the foundation of tensor analysis in some detail for the convenience of readers not familiar enough with this important subject. The present chapter is the presentation of the foundation of tensor analysis in some detail for the convenience of readers not familiar enough with this important subject.
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Notes
- 1.
In tensor analysis, designation of a vector or a tensor by the use of a bold-face letter, just as in the left-hand side of Eq. (12.7a), is called direct notation (or absolute notation). On the other hand, the designation of a vector or a tensor in terms of their components, as in the right-hand side of the second line of Eq. (12.7a) is said to be index notation.
- 2.
In the case of a scalar product between a tensor and a vector, or between two tensors, the dot (·) representing the scalar product is usually omitted by convention. Namely we do not write \({\textbf{\textit{S}}} \cdot {\textbf{\textit{u}}}\) nor \({\textbf{\textit{S}}} \cdot {\textbf{\textit{T}}}\).
- 3.
3A coordinate system based on an orthonormal basis is said to be an orthonormal coordinate system, or a Cartesian coordinate system.
- 4.
E i and \(G_{ij} \;\left(i,\, j = 1,\,2,\,3\right)\) denote Young’s modulus in x i -direction and the shear modulus in \(x_i x_j\)-plane. The symbol \(\nu _{ij} = - \left(\varepsilon _j /\varepsilon _i \right)\) is Poisson’s ratio of the strain in x j -direction to that of x i -direction.
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Murakami, S. (2012). Foundations of Tensor Analysis – Tensor Algebra and Tensor Calculus. In: Continuum Damage Mechanics. Solid Mechanics and Its Applications, vol 185. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2666-6_12
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DOI: https://doi.org/10.1007/978-94-007-2666-6_12
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