Properties for Design of Composite Structures. Neil McCartney

Properties for Design of Composite Structures

href="#ulink_2a4bcde2-c305-5db1-bdb6-95d1b181fc94">2.22) that

(2.25)

where use has been made of the following identity

nabla period left-parenthesis zero width space zero width space phi v right-parenthesis identical-to phi nabla period v plus v period nabla phi period (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.

2.6 Continuity Equation

Consider at time t a moving sample of material occupying a fixed volume V bounded by a closed surface S. In the absence of mass source and sink terms, the total mass of material within the region V is fixed so that on setting ϕ = ρ in (2.27) the global form of the mass balance equation for the medium may be written as

integral Underscript upper V Endscripts StartFraction partial-differential rho Over partial-differential t EndFraction normal d upper V equals minus integral Underscript normal upper S Endscripts rho v period n normal d upper S comma (2.28)

where ρ is the mass density. The term on the left-hand side of (2.28) is the rate of change of the total mass in the fixed region V. The term on the right-hand side is the rate at which the mass of the medium is transported across the surface S into the region V. On using the divergence theorem, the mass balance equation may be written as

integral Underscript normal upper V Endscripts StartFraction partial-differential rho Over partial-differential t EndFraction normal d upper V equals minus integral Underscript normal upper V Endscripts nabla period left-parenthesis rho v right-parenthesis normal d upper V period (2.29)

As (2.29) must be valid for any region V of the system, the following local form of the mass balance equation for the medium must be satisfied at all points in the system for all times t

StartFraction partial-differential rho zero width space Over partial-differential t EndFraction equals minus nabla period left-parenthesis zero width space zero width space rho v right-parenthesis period (2.30)

On using (2.22) and the identity (2.26), the continuity equation (2.30) may be written in the equivalent form

StartFraction upper D rho zero width space Over upper D t EndFraction equals minus rho nabla period v period (2.31)

2.7 Equations of Motion and Equilibrium

The global form of the linear momentum balance equation for a fixed region V bounded by the closed surface S having outward unit normal n is written as

StartFraction normal d Over normal d t EndFraction integral Underscript normal upper V Endscripts rho v normal d upper V equals integral Underscript normal upper S Endscripts n period sigma d upper S plus integral Underscript upper V Endscripts rho b normal d upper V comma (2.32)

where σ is the stress tensor and where b is the body force per unit mass acting on the medium. Such body forces usually arise from the effects of gravity. On using (2.27) and the divergence theorem, the linear momentum balance equation (2.32) may be written as

(2.33)

As relation (2.33) must be satisfied for any region V of the system, it follows that the local form of the linear momentum balance equation has the form

StartFraction partial-differential left-parenthesis rho v right-parenthesis Over partial-differential t EndFraction plus nabla period left-parenthesis rho v v right-parenthesis equals nabla period sigma plus rho zero width space b period (2.34)

nabla period left-parenthesis rho v v right-parenthesis identical-to rho v nabla period v plus v period nabla left-parenthesis rho v right-parenthesis comma (2.35)

it follows from (2.34) on using (2.22) that

rho StartFraction normal upper D v Over normal upper D t EndFraction plus v StartFraction upper D rho Over upper D t EndFraction plus rho v nabla period v equals nabla period sigma plus rho b period (2.36)

On using (2.31), relation (2.36) reduces to the well-known equation of motion

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