Properties for Design of Composite Structures. Neil McCartney

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СКАЧАТЬ V Endscripts StartFraction partial-differential phi Over partial-differential t EndFraction normal d upper V plus integral Underscript upper S Endscripts n period v phi normal d upper S period"/>(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

(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

(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

(2.30)

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

(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

(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

(2.34)

      As

      (2.35)

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

(2.36)

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

(2.37)

      which must be satisfied at every point in the medium for all times t > 0.

It is worth noting that by considering the balance of angular momentum it can be shown that in the absence of couple stresses, the stress tensor σ must be symmetric.

      2.8 Energy Balance Equation

      The energy balance equation is derived from the first law of thermodynamics which requires that, for a fixed region V bounded by a closed surface S, the rate of change of the sum of the internal energy and kinetic energy is balanced by the sum of the heat flowing across the external surface S, and the rate of working of the external tractions acting on S and of the body force acting in V. There are two types of stored energy that must be considered. The first is the internal energy that accounts for the strain energy stored owing to elastic deformation and the energy of the thermal agitations of atoms in the solid. СКАЧАТЬ