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Unformatted text preview: Section 2.13 Solid Mechanics Part III Kelly 308 Variation and Linearisation of Kinematic Tensors 2.13.1 The Variation of Kinematic Tensors The Variation In this section is reviewed the concept of the variation, introduced in Part I, §5.5. The variation is defined as follows: consider a function ) ( x u , with ) ( x u * a second function which is at most infinitesimally different from ) ( x u at every point x , Fig. 2.13.1 Figure 2.13.1: the variation Then define ) ( ) ( x u x u u − = * δ The Variation (2.13.1) The operator δ is called the variation symbol and u δ is called the variation of ) ( x u . The variation of ) ( x u is understood to represent an infinitesimal change in the function at x . Note from the figure that a variation u δ of a function u is different to a differential u d . The ordinary differentiation gives a measure of the change of a function resulting from a specified change in the independent variable (in this case x ). Also, note that the independent variable does not participate in the variation process; the variation operator imparts an infinitesimal change to the function u at some fixed x – formally, one can write this as = x δ . The Commutative Properties of the variation operator (1) x u u x d d d d δ δ = ( 2 . 1 3 . 2 ) ) ( x u δ ) ( x u x d u d x ) ( * x u Section 2.13 Solid Mechanics Part III Kelly 309 Proof : ( ) ) ( ) * ( * * x u x x u u x u x u x u x u x u δ δ d d d d d d d d d d d d d d = − = − = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (2) ∫ ∫ = 2 1 2 1 ) ( ) ( x x x x d d x x u x x u δ δ ( 2 . 1 3 . 3 ) Proof : [ ] ∫ ∫ ∫ ∫ ∫ = − = − = 2 1 2 1 2 1 2 1 2 1 ) ( ) ( ) ( * ) ( ) ( * ) ( x x x x x x x x x x x x u x x u x u x x u x x u x x u d d d d d δ δ Variation of a Function Consider A , a scalar-, vector-, or tensor-valued function of u . The value of A at u u δ + , where u δ is a variation of u is, as in, for example, 1.15.27, ] [ ) ( ) ( u A u A u u A u δ δ ∂ + ≈ + (2.13.4) The directional derivative in this context is also denoted by ( ) u u A δ δ , and is called the variation of A : ( ) ( ) u u A u A u u A u εδ ε δ δ δ ε + = ∂ ≡ = ] [ , d d (2.13.5) The variation of...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.
- Fall '11