Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 18

The concept of a material time derivative (often named the substantive derivative) associated with the motion of the material is fundamental to the mechanics of continuous media. If ϕ is any scalar, vector or tensor quantity such that ϕ(x,t)≡ϕ¯(x¯,t), then its material time derivative is defined by

(2.21)

On using (2.20) the material time derivative may then be written as

(2.22)

where ∇ denotes the gradient with respect to the coordinates x.

Consider now a region V of the system bounded by the closed surface S enclosing a sample of the medium which is moving such that the velocity distribution at time t is denoted by v(x,t). It follows that, for any extensive property ϕ(x,t) of the system, the material time derivative (associated with the mean motion) of the integral of ϕ(x,t) over the region fixed region V may be written as

(2.23)

From reference [1, Equation (2.4.9)], for example,

(2.24)

On substituting (2.24) into (2.23) it follows on using (2.22) that

(2.25)

where use has been made of the following identity

(2.26)

On using the divergence theorem, the identity (2.25) may then be written as

(2.27)

The first term on the right-hand side accounts for any changes of the property ϕ locally at points within the region V, whereas the second term accounts for the mean advection of the property across the bounding surface S where n is the outward unit normal to the surface S bounding the region V. The important identities (2.22) and (2.27) are used repeatedly in the following analysis.