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By Tadmor E.B, Miller R.E., Elliott R.S.
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Additional resources for Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations
Sample text
There can be as many free indices as necessary. For example, the expression Dij k xk = Aij contains the two free indices i and j and therefore represents n2d equations. 4 Matrix notation Indicial operations involving tensors of rank two or less can be represented as matrix operations. For example, the product Aij xj can be expressed as a matrix multiplication. For nd = 3 we have ⎤⎡ ⎤ ⎡ x1 A11 A12 A13 ⎦ ⎣ ⎣ Aij xj = Ax = A21 A22 A23 x2 ⎦ . A31 A32 A33 x3 We use a sans serif font to denote matrices to distinguish them from tensors.
Ai , . . , an ], ∀ ai , ai ∈ V and ∀ λ, μ ∈ R. We now turn to the definition of the Euclidean space. Euclidean space The real coordinate space Rn d is an nd -dimensional vector space defined over the field of real numbers. A vector in Rn d is represented by a set of nd real components relative to a given basis. Thus for a ∈ Rn d we have a = (a1 , . . , an d ), where ai ∈ R. Addition and multiplication are defined for Rn d in terms of the corresponding operations familiar to us from the algebra of real numbers: 1.
3 What is a tensor? 27 A set satisfying these conditions is called a Euclidean point space. A position vector x for a point x is defined by singling out one of the points as the origin o and writing: x ≡ v(x, o). 18) imply that every point x in E is uniquely associated with a vector x in Rn d . The vector connecting two points is given by x − y = v(x, o) − v(y, o). The distance between two points and the angles formed by three points can be computed using the norm and inner product of the corresponding translation space.