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Unformatted text preview: Section 1.14 Solid Mechanics Part III Kelly 115 1.14 Tensor Calculus I: Tensor Fields In this section, the concepts from the calculus of vectors are generalised to the calculus of higherorder tensors. 1.14.1 Tensorvalued Functions Tensorvalued functions of a scalar The most basic type of calculus is that of tensorvalued functions of a scalar, for example the timedependent stress at a point, ) ( t S S = . If a tensor T depends on a scalar t , then the derivative is defined in the usual way, t t t t dt d t Δ − Δ + = → Δ ) ( ) ( lim T T T , which turns out to be j i ij dt dT dt d e e T ⊗ = (1.14.1) The derivative is also a tensor and the usual rules of differentiation apply, ( ) ( ) ( ) ( ) ( ) T T ) ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + = + = + = + = + dt d dt d dt d dt d dt d dt d dt d dt d dt d dt d t dt d dt d dt d dt d T T B T B T TB a T a T Ta T T T B T B T α α α For example, consider the time derivative of T QQ , where Q is orthogonal. By the product rule, using I QQ = T , ( ) Q Q Q Q Q Q Q Q QQ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = + = T T T T T dt d dt d dt d dt d dt d Thus, using Eqn. 1.10.3e ( ) T T T T Q Q Q Q Q Q & & & − = − = (1.14.2) Section 1.14 Solid Mechanics Part III Kelly 116 which shows that T Q Q & is a skewsymmetric tensor. 1.14.2 Vector Fields The gradient of a scalar field and the divergence and curl of vector fields have been seen in §1.6. Other important quantities are the gradient of vectors and higher order tensors and the divergence of higher order tensors. First, the gradient of a vector field is introduced. The Gradient of a Vector Field The gradient of a vector field is defined to be the secondorder tensor j i j i j j x a x e e e a a ⊗ ∂ ∂ = ⊗ ∂ ∂ ≡ grad Gradient of a Vector Field (1.14.3) In matrix notation, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 grad x a x a x a x a x a x a x a x a x a a (1.14.4) One then has ( ) ) ( ) ( grad x a x x a a e e e e x a d d d dx x a dx x a d i j j i k k j i j i − + = = ∂ ∂ = ⊗ ∂ ∂ = (1.14.5) which is analogous to Eqn 1.6.7 for the gradient of a scalar field. As with the gradient of a scalar field, if one writes x d as e x d , where e is a unit vector, then direction in grad e a e a ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = dx d (1.14.6) Thus the gradient of a vector field a is a secondorder tensor which transforms a unit vector into a vector describing the gradient of a in that direction. For a space curve parameterised by s , one has Section 1.14 Solid Mechanics Part III Kelly 117 ( ) τ a τ e a e τ a a a grad = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⊗ ∂ ∂ = ⋅ ∂ ∂ = ∂ ∂ = i i i i i i x x ds dx x ds d where τ is a tangent vector to C (see §1.6.2)....
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 Fall '11
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 Vector Calculus, Tensor, Vector field, Tensor field

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